Application of Derivatives
153 JEE Maths previous year questions on Application of Derivatives — options free on every question; 15 include the answer & explanation free, the rest unlock with PYQ Pass.
Let \((2,3)\) be the largest open interval in which the function \(f(x)=2 \log _{\mathrm{e}}(x-2)-x^2+a x+1\) is strictly increasing and \((b, c)\) be the largest open interval, in which the function \(\mathrm{g}(x)=(x-1)^3(x+2-\mathrm{a})^2\) is strictly decreasing. Then \(100(a+b-c)\) is equal to :
[JEE Main 2025, 24 Jan (Shift 2)]
360
\(\text{Given,}\\ \mathrm{f}\left(\mathrm{x}\right)=2{\log }_{\mathrm{e}}\left(\mathrm{x}-2\right)-{\mathrm{x}}^{2}+\mathrm{ax}+1\\ \text{On differentiating both sides, we get}\\ \mathrm{f}'\left(\mathrm{x}\right)=\frac{2}{\mathrm{x}-2}-2\mathrm{x}+\mathrm{a}\geq 0\left[\text{for increasing}f\left(x\right)\right]\\ \text{Again,differentiating both sides, we get}\\ \mathrm{f}"\left(\mathrm{x}\right)=\frac{-2}{(\mathrm{x}-2{)}^{2}}-2<0\\ f'\left(x\right)\text{ is decreasing function }\\ {\mathrm{f}}^{'}\left(3\right)\geq 0\\ 2-6+\mathrm{a}\geq 0\\ \mathrm{a}\geq 4\\ {\mathrm{a}}_{\min }=4\\ \text{Now, }\\ \mathrm{g}\left(\mathrm{x}\right)={\left(\mathrm{x}-1\right)}^{3}{\left(\mathrm{x}+2-\mathrm{a}\right)}^{2}\\ \mathrm{g}\left(\mathrm{x}\right)={\left(\mathrm{x}-1\right)}^{3}{\left(\mathrm{x}-2\right)}^{2}\\ \text{On differentiating, we get}\\ {\mathrm{g}}^{'}\left(\mathrm{x}\right)={\left(x-1\right)}^{3}2\left(x-2\right)+{\left(\mathrm{x}-2\right)}^{2}3{\left(x-1\right)}^{2}\\ ={\left(x-1\right)}^{2}\left(\mathrm{x}-2\right)\left(2\mathrm{x}-2+3\mathrm{x}-6\right)\\ ={\left(x-1\right)}^{2}\left(\mathrm{x}-2\right)\left(5\mathrm{x}-8\right)<0\\ \mathrm{x}\in \left(\frac{8}{5},2\right)\\ \text{Hence, }\\ 100\left(\mathrm{a}+\mathrm{b}-\mathrm{c}\right)=100\left(4+\frac{8}{5}-2\right)=360\)
Let \(f:\to \mathrm{ℝ}\to (0,\infty )\) be strictly increasing function such that \(\lim _{x\to \infty }\frac{f(7x)}{f(x)}=1\). Then, the value of \(\lim _{x\to \infty }\left[\frac{f(5x)}{f(x)}-1\right]\) is equal to
[JEE Main 2024, 31 Jan (Shift 2)]
0
\(f\) is increasing function.
\(x<5x<7x\)
\(f(x) \(\frac{f(x)}{f(x)}<\frac{f(5x)}{f(x)}<\frac{f(7x)}{f(x)}\) \(\lim_{x\to\infty}\frac{f(x)}{f(x)}<\lim_{x\to\infty}\frac{f(5x)}{f(x)}<\lim_{x\to\infty}\frac{f(7x)}{f(x)}\) \(1<\lim_{x\to\infty}\frac{f(5x)}{f(x)}<1\Rightarrow \lim_{x\to\infty}\frac{f(5x)}{f(x)}=1\) \(\lim_{x\to\infty}\left(\frac{f(5x)}{f(x)}-1\right)=0\)
Let \(g(x)=3 f\left(\frac{x}{3}\right)+f(3-x) \forall x \in(0,3)\) and \(f^{\prime\prime}(x)>0 \forall x \in(0,3)\) then \(g(x)\) decreases in interval \((0, \alpha)\), then \(\alpha\) is (22 Jan, Shift I, Memory Based)
\(\frac{9}{4}\)
\(g(x)=3 f\left(\frac{x}{3}\right)+f(3-x) \text { and } f^{\prime \prime}(x)>0 \forall x \in(0,3)\)
\(\Rightarrow f^{\prime}(x)\) is increasing function
\(\begin{aligned}& g^{\prime}(x)=3 \times \frac{1}{3} \cdot f^{\prime}\left(\frac{x}{3}\right)-f^{\prime}(3-x) \\& =\mathrm{f}^{\prime}\left(\frac{\mathrm{x}}{3}\right)-\mathrm{f}^{\prime}(3-\mathrm{x})\end{aligned}\)
If g is decreasing in \((0, \alpha)\)
\(\begin{aligned}& \mathrm{g}^{\prime}(\mathrm{x})<0 \\& \mathrm{f}^{\prime}\left(\frac{\mathrm{x}}{3}\right)-\mathrm{f}^{\prime}(3-\mathrm{x})<0 \\& \mathrm{f}^{\prime}\left(\frac{\mathrm{x}}{3}\right)<\mathrm{f}^{\prime}(3-\mathrm{x}) \\& \Rightarrow \frac{\mathrm{x}}{3}<3-\mathrm{x} \\& \Rightarrow \mathrm{x}<\frac{9}{4}\end{aligned}\)
Let \(f(x)=\int_0^{x^2} \frac{t^2-8 t+15}{e^t} d t, x \in \mathbf{R}\). Then the numbers of local maximum and local minimum points of \(f\), respectively, are :
[JEE Main 2025, 22 Jan (Shift 2)]
2 and 3
\(f\left(x\right)={\int }_{0}^{{x}^{2}}\frac{{t}^{2}-8t+15}{{e}^{t}}dt,x\in R\\ {f}^{'}\left(x\right)=\left(\frac{{x}^{4}-8{x}^{2}+15}{{e}^{{x}^{2}}}\right)\left(2x\right)\\ =\frac{\left({x}^{2}-3\right)\left({x}^{2}-5\right)(2x)}{{\mathrm{e}}^{{\mathrm{x}}^{2}}}\\ =\frac{(x-\sqrt{3})(x+\sqrt{3})(x-\sqrt{5})(x+\sqrt{5})2x}{{e}^{{x}^{2}}}\)
using first derivative approach :-
\(\text{Maxima at }x\in {-\sqrt{3},\sqrt{3}}\\ \text{Minima at }\mathrm{x}\in {-\sqrt{5},0,\sqrt{5}}\)
number of local maxima points=\(2\)
number of local minima points=\(3\)
The shortest distance between the curves \({\mathrm{y}}^{2}=8\mathrm{x}\) and \({x}^{2}+{y}^{2}+12y+35=0\)is :
[JEE Main 2025, 3 Apr (Shift 2)]
\(2\sqrt{2}-1\)
Complete the square for the circle equation:
\({x}^{2}+{y}^{2}+12y+35=0\text{ }\\ ⟹\text{ }{x}^{2}+({y}^{2}+12y)+35=0\)
Complete the square for \(y\):
\({y}^{2}+12y=(y+6{)}^{2}−36\)
Substitute back:
\({x}^{2}+(y+6{)}^{2}−36+35=0\text{ }\\ ⟹\text{ }{x}^{2}+(y+6{)}^{2}=1\)
The circle has center \((0,−6)\) and radius \(r=1\).
The parabola \({y}^{2}=8x\) can be parameterized as \((\frac{{t}^{2}}{8},t)\) (where \(t\) is a parameter).
The distance \(D\) from the circle’s center \((0,−6)\) to a point \((\frac{{t}^{2}}{8},t)\) on the parabola is:
\(D=\sqrt{{(\frac{{t}^{2}}{8}−0)}^{2}+(t−(−6){)}^{2}}\\ =\sqrt{\frac{{t}^{4}}{64}+(t+6{)}^{2}}\)
To minimize \(D\), minimize \({D}^{2}\) (since the square root is monotonic):
Take the derivative of \(f(t)\) and set it to zero:
\({f}^{′}(t)=\frac{4{t}^{3}}{64}+2(t+6)=\frac{{t}^{3}}{16}+2t+12\)
Set \({f}^{′}(t)=0\):
\(\frac{{t}^{3}}{16}+2t+12=0\text{ }⟹\text{ }{t}^{3}+32t+192=0\)
By the Rational Root Theorem, \(t=−4\) is a root (verify: \((−4{)}^{3}+32(−4)+192\)\(=−64−128+192=0\)). Factor the cubic:
\({t}^{3}+32t+192=(t+4)({t}^{2}−4t+48)\)
The quadratic \({t}^{2}−4t+48\) has no real roots (discriminant \(−176<0\)), so the only real critical point is \(t=−4\).
Evaluate \(f(t)\) at \(t=−4\):
\(f(−4)=\frac{(−4{)}^{4}}{64}+(−4+6{)}^{2}\)\(=\frac{256}{64}+{2}^{2}=4+4=8\)
Thus, \(D=\sqrt{8}=2\sqrt{2}\).
The shortest distance between the circle and the parabola is the minimum distance from the center to the parabola minus the circle’s radius:
\(\text{Shortest distance}=2\sqrt{2}−1\)
Mathematics
Let \(g(x)=3 f\left(\frac{x}{3}\right)+f(3-x)\) and \(f^{\prime \prime}(x)>0\) for all \(x \in(0,3)\). If \(g\) is decreasing in \((0, \alpha)\) and increasing in \((\alpha, 3)\), then \(8 \alpha\) is :
[JEE Main 2024, 27 Jan (Shift 2)]
18
\(g(x) = 3f \left( \frac{x}{3} \right) + f(3 - x)\) and \(f''(x) > 0 \forall x \in (0,3)\)
\(\Rightarrow f'(x)\) is increasing function
\(g'(x) = 3 \times \frac{1}{3} f' \left( \frac{x}{3} \right) - f'(3 - x) = f' \left( \frac{x}{3} \right) - f'(3 - x)\)
If g is decreasing in \((0, \alpha)\)
\(g'(x) < 0\)
\(= f' \left( \frac{x}{3} \right) - f'(3-x) < 0\)
\(\Rightarrow f' \left( \frac{x}{3} \right) < f'(3-x)\)
\(\Rightarrow \frac{x}{3} < 3-x \Rightarrow x < \frac{9}{4}\)
Therefore \(\alpha = \frac{9}{4}\)
Then \(8\alpha = 8 \times \frac{9}{4} = 18\)
Mathematics
Let \(g(x)=3 f\left(\frac{x}{3}\right)+f(3-x)\) and \(f^{\prime \prime}(x)>0\) for all \(x \in(0,3)\). If \(g\) is decreasing in \((0, \alpha)\) and increasing in \((\alpha, 3)\), then \(8 \alpha\) is :
[JEE Main 2024, 27 Jan (Shift 2)]
18
\(g(x) = 3f \left( \frac{x}{3} \right) + f(3 - x)\) and \(f''(x) > 0 \forall x \in (0,3)\)
\(\Rightarrow f'(x)\) is increasing function
\(g'(x) = 3 \times \frac{1}{3} f' \left( \frac{x}{3} \right) - f'(3 - x) = f' \left( \frac{x}{3} \right) - f'(3 - x)\)
If g is decreasing in \((0, \alpha)\)
\(g'(x) < 0\)
\(= f' \left( \frac{x}{3} \right) - f'(3-x) < 0\)
\(\Rightarrow f' \left( \frac{x}{3} \right) < f'(3-x)\)
\(\Rightarrow \frac{x}{3} < 3-x \Rightarrow x < \frac{9}{4}\)
Therefore \(\alpha = \frac{9}{4}\)
Then \(8\alpha = 8 \times \frac{9}{4} = 18\)
If the function \(f(x)=2{x}^{3}-9a{x}^{2}+12{a}^{2}x+1\), where a \(>0\), attains its local maximum and local minimum values at p and q , respectively, such that \({\mathrm{p}}^{2}=\mathrm{q}\), then \(f(3)\) is equal to:
[JEE Main 2025, 2 Apr (Shift 1)]
\(37\)
The first derivative of \(f(x)\) is:
\({f}^{′}(x)=6{x}^{2}−18ax+12{a}^{2}\)
Factor the derivative:
\({f}^{′}(x)=6({x}^{2}−3ax+2{a}^{2})=6(x−a)(x−2a)\)
Setting \({f}^{′}(x)=0\) gives critical points \(x=a\) and \(x=2a\).
Analyze the sign of \({f}^{′}(x)\):
- For \(x
- For \(a
- For \(x>2a\), \({f}^{′}(x)>0\) (function increasing).
Thus, \(x=a\) is a local maximum ( \(p=a\) ) and \(x=2a\) is a local minimum ( \(q=2a\) ).
Given \({p}^{2}=q\), substitute \(p=a\) and \(q=2a\):
\({a}^{2}=2a\)
Since \(a>0\), divide both sides by \(a\):
\(a=2\)
Substitute \(a=2\) into \(f(x)\):
\(f(x)=2{x}^{3}−9(2){x}^{2}+12(2{)}^{2}x+1=2{x}^{3}−18{x}^{2}+48x+1\)
Now, evaluate \(f(3)\):
\(f(3)=2(3{)}^{3}−18(3{)}^{2}+48(3)+1\)
Thus, \(f(3)=37\)
Let the function \(\mathrm{f}(\mathrm{x})=\frac{\mathrm{x}}{3}+\frac{3}{\mathrm{x}}+3,\mathrm{x}\neq 0\) be strictly increasing in \(\left(-\infty ,{\alpha }_{1}\right)\cup \left({\alpha }_{2},\infty \right)\) and strictly decreasing in \(\left({\alpha }_{3},{\alpha }_{4}\right)\cup \left({\alpha }_{4},{\alpha }_{5}\right)\). Then \(\sum _{i=1}^{5}{\alpha }_{i}^{2}\) is equal to :-
[JEE Main 2025, 8 Apr (Shift 1)]
\(36\)
Sol:
\(\begin{matrix}f'(x)=\frac{1}{3}−\frac{3}{{x}^{2}} \\ =\frac{{x}^{2}−9}{3{x}^{2}}=\frac{(x+3)(x−3)}{3{x}^{2}}\end{matrix}\)
For strictly increasing \({\mathrm{f}}^{'}(\mathrm{x})>0\Rightarrow \mathrm{x}\in (-\infty ,-3)\cup (3,\infty )\)
For strictly decreasing \({\mathrm{f}}^{'}(\mathrm{x})<0\Rightarrow \mathrm{x}\in (-3,0)\cup (0,3)\)
\(∴{\alpha }_{1}=−3,{\alpha }_{2}=3\ {\alpha }_{3}=−3,{\alpha }_{4}=0\ {\alpha }_{5}=3\)
\(\sum _{i=1}^{5}{\alpha }_{i}^{2}=36\)
Let the function \(\mathrm{f}(\mathrm{x})=\frac{\mathrm{x}}{3}+\frac{3}{\mathrm{x}}+3,\mathrm{x}\neq 0\) be strictly increasing in \(\left(-\infty ,{\alpha }_{1}\right)\cup \left({\alpha }_{2},\infty \right)\) and strictly decreasing in \(\left({\alpha }_{3},{\alpha }_{4}\right)\cup \left({\alpha }_{4},{\alpha }_{5}\right)\). Then \(\sum _{i=1}^{5}{\alpha }_{i}^{2}\) is equal to :-
[JEE Main 2025, 8 Apr (Shift 1)]
\(36\)
\(f(x)=\frac{x}{3}+\frac{3}{x}+3,\ x\neq 0\)
\(f'(x)=\frac{1}{3}-\frac{3}{x^2}=0\Rightarrow x=\pm 3\)
\(f'(x)=\frac{x^2-3}{3x^2}\)
\(f'(x)>0\ \forall(-\infty,-3)\cup(3,\infty)\rightarrow\) increasing
\(f'(x)<0\ \forall(-3,0)\cup(0,3)\rightarrow\) decreasing
\(\sum_{i=1}^{5}\alpha_i^2=(-3)^2+(3)^2+(-3)^2+(0)^2+(3)^2=36\)
Let \(f(x)=\int_0^{x^2} \frac{t^2-8 t+15}{e^t} d t, x \in \mathbf{R}\). Then the numbers of local maximum and local minimum points of \(f\), respectively, are :
[JEE Main 2025, 22 Jan (Shift 2)]
2 and 3
\(f(x)={\int }_{0}^{{x}^{2}}\frac{{t}^{2}-8t+15}{{e}^{t}}dt,x\in R\\ \text{using lebnitz rule:}\\ {f}^{'}(x)=\left(\frac{{x}^{4}-8{x}^{2}+15}{{e}^{{x}^{2}}}\right)(2x)\\ =\frac{\left({x}^{2}-3\right)\left({x}^{2}-5\right)(2x)}{{\mathrm{e}}^{{\mathrm{x}}^{2}}}\\ =\frac{(x-\sqrt{3})(x+\sqrt{3})(x-\sqrt{5})(x+\sqrt{5})2x}{{e}^{{x}^{2}}}\)
using first derivative approach :-
\(\text{ Maxima at }x\in {-\sqrt{3},\sqrt{3}}\\ \text{ Minima at }\mathrm{x}\in {-\sqrt{5},0,\sqrt{5}}\)
number of local maxima points=\(2\)
number of local minima points=\(3\)
Consider the following three statements for the function \(f:\left(0,∞\right)\to ℝ\) defined by \(f\left(x\right)=\text{ }∣{\text{log}}_{e}x∣−|x−1∣:\)
(I) \(f\) is differentiable at all \(x>0\).
(II) \(f\) is increasing in \(\left(0,1\right)\).
(III) \(f\) is decreasing in \(\left(1,\infty \right)\). Then.
[JEE Main 2026, 24 Jan (Shift 2)]
Only (I) and (III) are TRUE.
\(f(x)=|lnx|−|x−1|\)
\(=\left\{\begin{matrix}lnx−(x−1) & x\geq 1 \\ −lnx+(x−1) & 0 \(=\left\{\begin{matrix}lnx−x+1 & x\geq 1 \\ −lnx+x−1 & 0 \({f}^{'}\left(x\right)=\left\{\begin{matrix}\frac{1}{x}-1 & x\geq 1 \\ -\frac{1}{x}+1 & 0 \({f}^{'}\left({1}^{+}\right)={f}^{'}\left({1}^{−}\right)=0\) \(\Rightarrow f(x)\) is differentiable \(\forall x>0\). \(f'(x)<0\forall x>1\\ f'(x)<0\forall 0 \(\Rightarrow f(x)\) is decreasing \(\forall x\in (0,\infty )\).
Consider the function \(f:\left(0,\infty \right)\to \left(−\infty ,\infty \right)\) given by \(f\left(x\right)=\sqrt{x}{\text{log}}_{\text{e}}\left(x\right)−x+1\)
Then which one of the following statements is TRUE ?
[JEE Advanced 2026]
The function \(f\) has NEITHER a point of local maximum NOR a point of local minimum in the interval \(\left(0,\infty \right)\)
Given \(f(x)=\sqrt{x}\ln x-x+1\), where \(x>0\).
\(f'(x)=\dfrac{1}{2\sqrt{x}}\ln x+\sqrt{x}\cdot\dfrac1x-1\)
\(f'(x)=\dfrac{\ln x}{2\sqrt{x}}+\dfrac1{\sqrt{x}}-1=\dfrac{\ln x+2}{2\sqrt{x}}-1\)
\(f'(x)=\dfrac{\ln x+2-2\sqrt{x}}{2\sqrt{x}}\)
Put \(t=\sqrt{x}\), so \(x=t^2\) and \(\ln x=2\ln t\).
\(f'(x)=\dfrac{2\ln t+2-2t}{2t}=\dfrac{\ln t+1-t}{t}\)
Now \(\ln t\le t-1\) for \(t>0\).
So \(\ln t+1-t\le0\), hence \(f'(x)\le0\).
Equality occurs only when \(t=1\), i.e. \(x=1\).
Therefore \(f\) is decreasing on \((0,\infty)\), but derivative becomes zero only at one point.
Now check option \(A\):
\(f''(x)=\dfrac{-\ln x}{4x^{3/2}}\)
For \(0
Thus \(f'\) is increasing on \((0,1)\), not decreasing. So option \(A\) is false.
Since \(f'(x)\le0\) on \((0,\infty)\), \(f\) cannot have a local maximum or local minimum. So options \(B\) and \(C\) are false.
Hence option \(D\) is true.
Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a differentiable function such that \(f\left(\frac{x+y}{3}\right)=\frac{f(x)+f(y)}{3}\) for all \(x, y \in \mathbb{R}\), and \(f^{\prime}(0)=3\). Then the minimum value of the function \(g(x)=3+e^x f(x)\) is:
[JEE Main 2026, 5 Apr (Shift 1)]
\(3\left(\frac{e-1}{e}\right)\)
\(\mathrm{f}\left(\frac{\mathrm{x}+\mathrm{y}}{3}\right)=\frac{\mathrm{f}(\mathrm{x})+\mathrm{f}(\mathrm{y})}{3}\)
Put \(\mathrm{x}=\mathrm{y}=0\)
\( f(0)=\frac{2 f(0)}{3}\)
\(\Rightarrow f(0)=0 \ldots(1) \)
\(\mathrm{f}^{\prime}\left(\frac{\mathrm{x}+\mathrm{y}}{3}\right) \cdot \frac{1}{3}=\frac{1}{3} \mathrm{f}^{\prime}(\mathrm{x}) \)
Put \( \mathrm{x}=0 \)
\( f^{\prime}\left(\frac{\mathrm{y}}{3}\right) \frac{1}{3}=\frac{1}{3} \times 3\)
\( \mathrm{f}^{\prime}\left(\frac{\mathrm{y}}{3}\right)=3 \)
Put \( \mathrm{y}=3 \mathrm{x}\)
\( \mathrm{f}^{\prime}(\mathrm{x})=3\)
Integrate both sides:
\(f(x)=3 x+C\), and from \(f(0)=0, C=0\), so \(f(x)=3 x\)
Now, \( \mathrm{g}(\mathrm{x})=3+\mathrm{e}^{\mathrm{x}} \cdot 3 \mathrm{x}\)
\( \mathrm{g}^{\prime}(\mathrm{x})=3\left[\mathrm{e}^{\mathrm{x}}+\mathrm{x} \cdot \mathrm{e}^{\mathrm{x}}\right]\)
\(=3 \mathrm{e}^{\mathrm{x}}(\mathrm{x}+1)\)
\( \mathrm{g}^{\prime}(\mathrm{x})=0\), at \( \mathrm{x}=-1\)
Now, \(g^{\prime \prime}(x)=3 e^x(2+x)\)
\(g^{\prime \prime}(-1)=3 e^{-1}(1)=\frac{3}{e}>0 \) (Local Minimum)
\((\mathrm{~g}(\mathrm{x}))_{\min }=3+\mathrm{e}^{-1}(-3)\)
\(=3\left[1-\frac{1}{\mathrm{e}}\right]=\frac{3(\mathrm{e}-1)}{\mathrm{e}}\)
The number of critical points of the function \(f(x)=(x-2)^{2 / 3}(2 x+1)\) is
[JEE Main 2024, 8 Apr (Shift 1)]
2
\(f(x)=(x-2)^{\frac{2}{3}}(2x+1)\)
\(f'(x)=\frac{2}{3}(x-2)^{-\frac{1}{3}}(2x+1)+(x-2)^{\frac{2}{3}}\)
\(f'(x)=2\times \frac{(2x+1)+(x-2)}{3(x-2)^{\frac{1}{3}}}\)
\(\frac{3x-1}{(x-2)^{\frac{1}{3}}}=0\)
Critical points \(x=\frac{1}{3}\) and \(x=2\)
The interval in which the function \(f(x)={x}^{x},x>0\), is strictly increasing is
[JEE Main 2024, 6 Apr (Shift 1)]
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Let \(x=-1\) and \(x=2\) be the critical points of the function \(\mathrm{f}(\mathrm{x})={\mathrm{x}}^{3}+{\mathrm{ax}}^{2}+{\mathrm{blog}}_{\mathrm{e}}|\mathrm{x}|+1,\mathrm{x}\neq 0\). Let m and M respectively be the absolute minimum and the absolute maximum values of f in the interval \(\left[-2,-\frac{1}{2}\right]\). Then \(|\mathrm{M}+m|\) is equal to ( Take \({\log }_{\mathrm{e}}2=0.7\)):
[JEE Main 2025, 7 Apr (Shift 1)]
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The maximum area of a triangle whose one vertex is at \((0,0)\) and the other two vertices lie on the curve \(y=-2 x^2+54\) at points \((x, y)\) and \((-x, y)\), where \(y>0\), is :
[JEE Main 2024, 30 Jan (Shift 1)]
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The maximum area of a triangle whose one vertex is at \((0,0)\) and the other two vertices lie on the curve \(y=-2 x^2+54\) at points \((x, y)\) and \((-x, y)\), where \(y>0\), is :
[JEE Main 2024, 30 Jan (Shift 1)]
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Let \(x=-1\) and \(x=2\) be the critical points of the function \(f(x)={x}^{3}+a{x}^{2}+{\mathrm{blog}}_{\mathrm{e}}|x|+1,x\neq 0\). Let \(m\) and \(M\) respectively be the absolute minimum and the absolute maximum values of \(f\) in the interval \(\left[-2,-\frac{1}{2}\right]\). Then \(|M+m|\) is equal to ( Take \({\log }_{\mathrm{e}}2=0.7\)):
[JEE Main 2025, 7 Apr (Shift 1)]
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If the function \(f(x)=2{x}^{3}-9a{x}^{2}+12{a}^{2}x+1,a>0\) has a local maximum at \(x=\alpha\) and a local minimum at \(x={\alpha }^{2}\), then \(\alpha\) and \({\alpha }^{2}\) are the roots of the equation :
[JEE Main 2024, 8 Apr (Shift 2)]
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Let \(\frac{{x}^{2}}{f\left({a}^{2}+7a+3\right)}+\frac{{y}^{2}}{f(3a+15)}=1\) represent an ellipse with major axis along \(y\)-axis, where \(f\) is a strictly decreasing positive function on \(R\). If the set of all possible values of \(a\) is \(R-[\alpha ,\beta ]\), then \({\alpha }^{2}+{\beta }^{2}\) is equal to:
[JEE Main 2026, 8 Apr (Shift 2)]
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\(\max _{0\leq \mathrm{x}\leq \pi }\left(16\sin \left(\frac{\mathrm{x}}{2}\right){\cos }^{3}\left(\frac{\mathrm{x}}{2}\right)\right)\) is equal to:
[JEE Main 2026, 4 Apr (Shift 2)]
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If the function \(f:(-\infty ,-1]\to (a,b]\) defined by \(f\left(x\right)={e}^{{x}^{3}-3x+1}\) is one - one and onto, then the distance of the point \(P(2b+4,a+2)\) from the line \(x+{e}^{-3}y=4\) is :
[JEE Main 2024, 31 Jan (Shift 2)]
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Let \((2,3)\) be the largest open interval in which the function \(f(x)=2 \log _{\mathrm{e}}(x-2)-x^2+a x+1\) is strictly increasing and \((b, c)\) be the largest open interval, in which the function \(\mathrm{g}(x)=(x-1)^3(x+2-\mathrm{a})^2\) is strictly decreasing. Then \(100(a+b-c)\) is equal to :
[JEE Main 2025, 24 Jan (Shift 2)]
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Let \(\mathrm{f}(\mathrm{x})=\left(2 \mathrm{x}^2-9 \mathrm{x}+11\right) \cdot \mathrm{e}^{\mathrm{x}},-\infty<\mathrm{x}<\infty\). Then \(f\) has :
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Let \(e\) be the base of natural logarithm and let \(f:{1,2,3,4}\to \left\{1,e,{e}^{2},{e}^{3}\right\}\) and \(g:\left\{1,e,{e}^{2},{e}^{3}\right\}\to \left\{1,\frac{1}{2},\frac{1}{3},\frac{1}{4}\right\}\) be two bijective functions such that \(f\) is strictly decreasing and \(g\) is strictly increasing. If \(ϕ\left(x\right)={\left[{f}^{-1}\left\{{g}^{-1}\left(\frac{1}{2}\right)\right\}\right]}^{x}\), then the area of the region \(R=\left\{(x,y):{x}^{2}\leq y\leq ϕ(x),0\leq x\leq 1\right\}\) is:
[JEE Main 2026, 6 Apr (Shift 1)]
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If the function \(f(x)=2{x}^{3}-9a{x}^{2}+12{a}^{2}x+1\), where \(a>0\), attains its local maximum and local minimum values at \(p\) and \(q,\) respectively, such that \({\mathrm{p}}^{2}=\mathrm{q}\), then \(f(3)\) is equal to:
[JEE Main 2025, 2 Apr (Shift 1)]
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Consider the function \(f:\left[\frac{1}{2}, 1\right] \rightarrow \mathbb{R}\) defined by \(f(x)=4 \sqrt{2} x^{3}-3 \sqrt{2} x-1\).
Consider the statements
(I) The curve \(y=f(x)\) intersects the \(x\)-axis exactly at one point.
(II) The curve \(y=f(x)\) intersects the \(x\)-axis at \(x=\cos \frac{\pi}{12}\).
Then
[JEE Main 2024, 29 Jan (Shift 1)]
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Consider the function \(f:\left[\frac{1}{2}, 1\right] \rightarrow \mathbb{R}\) defined by \(f(x)=4 \sqrt{2} x^{3}-3 \sqrt{2} x-1\).
Consider the statements
(I) The curve \(y=f(x)\) intersects the \(x\)-axis exactly at one point.
(II) The curve \(y=f(x)\) intersects the \(x\)-axis at \(x=\cos \frac{\pi}{12}\).
Then
[JEE Main 2024, 29 Jan (Shift 1)]
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Consider the function \(f(x)=|x-1| / x^2\), then \(f(x)\) is
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The function \(f(x)=\frac{x}{x^2-6 x-16}, x \in R -\{-2,8\}\)
[JEE Main 2024, 29 Jan (Shift 2)]
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The function \(f(x)=\frac{x}{x^2-6 x-16}, x \in R -\{-2,8\}\)
[JEE Main 2024, 29 Jan (Shift 2)]
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Let \(\mathrm{a}>0\). If the function \(\mathrm{f}(\mathrm{x})=6{\mathrm{x}}^{3}-45{\mathrm{ax}}^{2}+108{\mathrm{a}}^{2}\mathrm{x}+1\) attains its local maximum and minimum values at the points \({\mathrm{x}}_{1}\) and \({x}_{2}\) respectively such that \({\mathrm{x}}_{1}{\mathrm{x}}_{2}=54\), then \(\mathrm{a}+{\mathrm{x}}_{1}+{\mathrm{x}}_{2}\) is equal to :-
[JEE Main 2025, 4 Apr (Shift 2)]
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The function \( f(x)=3 x+\cos 3 x \) is:
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Let \(g : R \rightarrow R\) be a non constant twice differentiable function such that \(g ^{\prime}\left(\frac{1}{2}\right)= g ^{\prime}\left(\frac{3}{2}\right)\). If a real valued function \(f\) is defined as \(f(x)=\frac{1}{2}[g(x)+g(2-x)]\), then
[JEE Main 2024, 30 Jan (Shift 1)]
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Let \(g : R \rightarrow R\) be a non constant twice differentiable function such that \(g ^{\prime}\left(\frac{1}{2}\right)= g ^{\prime}\left(\frac{3}{2}\right)\). If a real valued function \(f\) is defined as \(f(x)=\frac{1}{2}[g(x)+g(2-x)]\), then
[JEE Main 2024, 30 Jan (Shift 1)]
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The function \( f(x)=3 x+\cos 3 x \) is:
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For the function \( f(x)=\sin x+3 x-\frac{2}{\pi}\left(x^2+x\right)\), where \(x \in\left[0, \frac{\pi}{2}\right]\), consider the following two statements :
(I) \(f\) is increasing in \(\left(0, \frac{\pi}{2}\right)\).
(II) \(f^{\prime}\) is decreasing in \(\left(0, \frac{\pi}{2}\right)\).
Between the above two statements,
[JEE Main 2024, 5 Apr (Shift 1)]
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Let \(g(x)=3 f\left(\frac{x}{3}\right)+f(3-x) \forall x \in(0,3)\) and \(f^{\prime\prime}(x)>0 \forall x \in(0,3)\) then \(g(x)\) decreases in interval \((0, \alpha)\), then \(\alpha\) is (22 Jan, Shift I, Memory Based)
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The smallest positive integral value of \(a\), for which all the roots of \({x}^{4}-a{x}^{2}+9=0\) are real and distinct, is equal to
[JEE Main 2026, 24 Jan (Shift 2)]
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Let \(\mathrm{a}>0\). If the function \(\mathrm{f}(\mathrm{x})=6{\mathrm{x}}^{3}-45{\mathrm{ax}}^{2}+108{\mathrm{a}}^{2}\mathrm{x}+1\) attains its local maximum and minimum values at the points \({\mathrm{x}}_{1}\) and \({x}_{2}\) respectively such that \({\mathrm{x}}_{1}{\mathrm{x}}_{2}=54\), then \(\mathrm{a}+{\mathrm{x}}_{1}+{\mathrm{x}}_{2}\) is equal to :-
[JEE Main 2025, 4 Apr (Shift 2)]
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Let \(f(x)=(x+3)^2(x-2)^3, x \in[-4,4]\). If \(M\) and \(m\) are the maximum and minimum values of \(f\), respectively in \([-4,4]\), then the value of \(M-m\) is
[JEE Main 2024, 30 Jan (Shift 2)]
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If \(5 f(x)+4 f\left(\frac{1}{x}\right)=x^2-2, \forall x \neq 0\) and \(y=9 x^2 f(x)\), then \(y\) is strictly increasing in :
[JEE Main 2024, 1 Feb (Shift 1)]
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For the function\(f(x)=(\cos x)-x+1,x\in \mathrm{ℝ}\), between the following two statements
\((S_1)\): \(f(x)=0\) for only one value of \(x\) in \([0, \pi]\).
\((S_2)\): \(f(x)\) is decreasing in \(\left[0,\frac{\pi }{2}\right]\) and increasing in \(\left[\frac{\pi }{2},\pi \right]\).
[JEE Main 2024, 8 Apr (Shift 1)]
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If the angle made by the tangent at the point ( \(x_0\),\(y_0\) ) on the curve
\(x=12(t+\sin t \cos t), y=12(1+\sin t)^2\), with \(0
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Let \(f(x)\) be a polynomial of degree \(5\), and have extrema at \(x=1\) and \(x=-1\). If \(\lim _{x\to 0}\left(\frac{f(x)}{{x}^{3}}\right)=-5\), then \(f(2)-f(-2)\) is equal to:
[JEE Main 2026, 2 Apr (Shift 2)]
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Consider the function \(f(x)=|x-1| / x^2\), then \(f(x)\) is
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A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is 1 cm , the ice-cream melts at the rate of \(81 \mathrm{~cm}^3 / \mathrm{min}\) and the thickness of the ice-cream layer decreases at the rate of \(\frac{1}{4 \pi} \mathrm{~cm} / \mathrm{min}\). The surface area (in \(\mathrm{cm}^2\) ) of the chocolate ball (without the ice-cream layer) is :
[JEE Main 2025, 23 Jan (Shift 2)]
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Let \(\mathrm{a}>0\). If the function \(\mathrm{f}(\mathrm{x})=6{\mathrm{x}}^{3}-45{\mathrm{ax}}^{2}+108{\mathrm{a}}^{2}\mathrm{x}+1\) attains its local maximum and minimum values at the points \({\mathrm{x}}_{1}\) and \({x}_{2}\) respectively such that \({\mathrm{x}}_{1}{\mathrm{x}}_{2}=54\), then \(\mathrm{a}+{\mathrm{x}}_{1}+{\mathrm{x}}_{2}\) is equal to :
[JEE Main 2025, 4 Apr (Shift 2)]
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If the angle made by the tangent at the point ( \(x_0\),\(y_0\) ) on the curve
\(x=12(t+\sin t \cos t), y=12(1+\sin t)^2\), with \(0
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Let a variable line of slope \(m>0\) passing through the point \((4,-9)\) intersect the coordinate axes at the points A and B . The minimum value of the sum of the distances of A and B from the origin is
[JEE Main 2024, 6 Apr (Shift 1)]
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If the curves \(x=y^4\) and \(x y=k\) cut at right angles, then \((4 k)^6\) is equal to.......
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Let \( f \) be a twice differentiable function on \( (1,6) \). If \( f(2)=8, f^{\prime}(2)=5, f^{\prime}(x) \geq 1 \) and \( f^{\prime \prime}(x) \geq 4 \), for all \( x \in(1,6) \), then
[JEE Main 2020, 4 Sep (Shift 1)]
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The maximum slope of the curve \(y=\frac{1}{2} x^4-5 x^3+18 x^2-\) \(19 x\) occurs at the point :
[JEE Main 2021, 26 Feb (Shift 1)]
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The function \(f(x)=\frac{4 x^3-3 x^2}{6}-2 \sin x+(2 x-1) \cos x\) :
[JEE Main 2021, 24 Feb (Shift 1)]
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Let \(A=\left[a_{i j}\right]\) be a \(3 \times 3\) matrix, where \(a_{i j}=\left\{\begin{array}{cc}1, & \text { if } i=j \\-x, & \text { if }|i-j|=1 \\2 x+1, & \text { otherwise }\end{array}\right.\)
Let a function \(f: R \rightarrow R\) be defined as \(f(x)=\operatorname{det}(A)\). Then the sum of maximum and minimum values of \(f\) on \(R\) is equal to:
[JEE Main 2021, 20 Jul (Shift 1)]
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If \(x=1\) is a critical point of the function \(f(x)=\left(3 x^2+a x-2-a\right)\) \(e^x\), then:
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If \(p(x)\) be a polynomial of degree three that has a local maximum value 8 at \(x = 1\) and a local minimum value 4 at \(x = 2\); then \(p(0)\) is equal to:
[JEE Main 2020, 2 Sep (Shift 1)]
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The function \(f(x)=x^3-6 x^2+a x+b\) is such that \(f(2)=f(4)=0\). [JEE Main 2021, 1 Sep (Shift 2)]
\(S_1\): there exists \(x_1, x_2 \in(2,4), x_1
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Let \(x=2\) be a local minima of the function \(f(x)=2 x^4-18 x^2+8 x+12, x \in(-4,4)\). If \(M\) is local maximum value of the function \(f\) in \((-4,4)\), then \(M=\)
[JEE Main 2023, 25 Jan (Shift 1)]
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The length of the perpendicular from the origin on the normal to the curve, \( x^{2}+2 x y-3 y^{2}=0 \) at the point \( (2,2) \) is
[JEE Main 2020, 8 Jan (Shift 2)]
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The curve given by \(x+y={e}^{xy}\) has a tangent parallel to \(Y\)-axis at the point
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Let \(f(x)\) be a cubic polynomial with \(f(1) = - 10,f( - 1) = 6\), and has a local minima at \(x = 1,f'(x)\) has a local minima at \(x = - 1\). Then \(f(3)\) is equal to
[JEE Main 2021, 31 Aug (Shift 2)]
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Let \(x=2\) be a local minima of the function \(f(x)=2 x^4-18 x^2\) \(+8 x+12, x \in(-4,4)\). If \(M\) is local maximum value of the function \(f\) in \((-4,4)\), then \(M=\)
[JEE Main 2023, 25 Jan (Shift 1)]
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The triangle of maximum area that can be inscribed in a given circle of radius ' \(r\) ' is:
[JEE Main 2021, 26 Feb (Shift 2)]
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The interval in which the function \(f(x)=\frac{4 x^2+1}{x}\) is decreasing is :
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Let us consider a curve, \(y=f(x)\) passing through the point \((-2,2)\) and the slope of the tangent to the curve at any point \((x, f(x))\) is given by \(f(x)+x f^{\prime}(x)=x^2\). Then:
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\[\max _{0 \leq x \leq \pi}\left\{x-2 \sin x \cos x+\frac{1}{3} \sin 3 x\right\}=\]
[JEE Main 2023, 13 Apr (Shift 1)]
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The value of \(a\in R\) for which the function \(\displaystyle f\left ( x \right )= \left ( 4a-3 \right )\left ( x+\log 5 \right )+2\left ( a-7 \right )\cot\left ( x/2 \right )\sin ^{2}\left ( x/2 \right )\) ,\(x\neq 2n\pi ,n\in N\) has critical points is
[JEE Main 2021, 16 Mar (Shift 1)]
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Let the normals at all the points on a given curve pass through a fixed point \((a, b)\). If the curve passes through \((3,-3)\) and \((4,-2 \sqrt{2})\) and given that \(a-2 \sqrt{2} b=3\), then \(\left(a^2+b^2+a b\right)\) is equal to ............
[JEE Main 2021, 26 Feb (Shift 2)]
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Let \( M \) and \( m \) respectively be the maximum and minimum values of the function \( f(x)=\tan ^{-1}(\sin x+\cos x) \) in \( \left[0, \frac{\pi}{2}\right] \). Then the value of \( \tan (M-m) \), is equal to
[JEE Main 2021, 27 Aug (Shift 2)]
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The equation of the normal to the curve \( y=(1+x)^{2 y}+\cos ^{2}\left(\sin ^{-1} x\right) \) at \( x=0 \) is :
[JEE Main 2020, 2 Sep (Shift 2)]
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A wire of length \(20 \mathrm{~m}\) is to be cut into two pieces. A piece of length \(l_1\) is bent to make a square of area \(A_1\) and the other piece of length \(l_2\) is made into a circle of area \(A_2\). If \(2 A_1+3 A_2\) is minimum then \(\left(\pi l_1\right): l_2\) is equal to:
[JEE Main 2023, 31 Jan (Shift 1)]
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The curve given by \(x+y=e^{x y}\) has a tangent parallel to the \(Y\)-axis at the point
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Let \(f\) be a real valued function, defined on \(R-\{-1,1\}\) and given by \(f(x)=3 \log _e\left|\frac{x-1}{x+1}\right|-\frac{2}{x-1}\). Then in which of the following intervals, function \(f(x)\) is increasing?
[JEE Main 2021, 16 Mar (Shift 2)]
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The value of \(c\) in the Lagrange's mean value theorem for the function \(f(x)=x^3-4 x^2+8 x+11\), when \(x \in[0,1]\) is:
[JEE Main 2020, 7 Jan (Shift 2)]
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Suppose that \(f(0)=-3\) and \(f^{\prime}(x) \leq 5\) for all values of \(x\). Then, the largest value which \(f(2)\) can attain is
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Let \(f:[2,4] \rightarrow \mathbb{R}\) be a differentiable function such that \(\left(x \log _e x\right)\) \(f^{\prime}(x)+\log _e x f(x)+f(x) \geq 1, x \in[2,4]\) with \(f(2)=\frac{1}{2}\) and \(f(4)=\frac{1}{4}\)
Consider the following two statements:
(A) : \(f(x) \leq 1\), for all \(x \in[2,4]\)
(B) : \(f(x) \geq \frac{1}{8}\), for all \(x \in[2,4]\) Then
[JEE Main 2023, 11 Apr (Shift 1)]
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Let \(f:(0,1) \rightarrow \mathrm{R}\) be a function defined by \(f(x)=\frac{1}{1-e^{-x}}\) and \(g(x)=(f(-x)-f(x))\).
Assertion (A): \(g\) is an increasing function in \((0,1)\)
Reason (R): \(g\) is one-one in \((0,1)\)
Then,
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The shortest distance between the line \(x-y=1\) and the curve \(x^2=2 y\) is:
[JEE Main 2021, 25 Feb (Shift 2)]
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If the equation of the normal to the curve \(y=\frac{x-a}{(x+b)(x-2)}\) at the point \((1,-3)\) is \(x-4 y=13\), then the value of \(a+b\) is equal to_________
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Let a function \( \mathrm{f}:[0,5] \rightarrow \mathrm{R} \) be continuous, \( \mathrm{f}(1)=3 \) and \(F\) be defined as: \( F(x)=\int_{1}^{x} t^{2} g(t) d t \), where \( g(t)=\int_{1}^{t} f(u) d u \). Then for the function \( F \), the point \( x=1 \) is :
[JEE Main 2020, 9 Jan (Shift 2)]
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The curve \( y(x)=a x^{3}+b x^{2}+c x+5 \) touches the \( x \)-axis at the point \( P(-2,0) \) and cuts the \( y \)-axis at the point \( Q\), where \(y^{\prime} \) is equal to 3 . Then the local maximum value of \( y(x) \) is
[JEE Main 2022, 25 Jul (Shift 1)]
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\(\max _{0 \leq x \leq \pi}\left\{x-2 \sin x \cos x+\frac{1}{3} \sin 3 x\right\}=\)
[JEE Main 2023, 13 Apr (Shift 1)]
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Let \(f(x)=3\left({\cos }^{4}x−2{\cos }^{2}x+1\right)+10{\sin }^{3}x+6{\sin }^{2}x−3,x.\in \left[−\frac{x}{6},\frac{x}{2}\right]\) . Then, \(f\) is:
[JEE Main 2021]
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A wire of length \(20 \mathrm{~m}\) is to be cut into two pieces one of the pieces is to be made into a square and the other into a regular hexagon. Then the length of the side (in meters) of the hexagon, so that the combined area of the square and the hexagon is minimum, is:
[JEE Main 2021, 27 Aug (Shift 1)]
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Let \(f(x)\) be a cubic polynomial with \(f(1) = - 10,f( - 1) = 6\), and has a local minima at \(x = 1,f'(x)\) has a local minima at \(x = - 1\). Then \(f(3)\) is equal to
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Let \(f(x)=2 x+\tan ^{-1} x\) and \(g(x)=\log _e\left(\sqrt{1+x^2}+x\right), x\) \(\in[0,3]\). Then
[JEE Main 2023, 1 Feb (Shift 1)]
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Let \( \mathrm{m} \) and \( \mathrm{M} \) be respectively the minimum and maximum values of \( \left|\begin{array}{ccc}\cos ^{2} x & 1+\sin ^{2} x & \sin 2 x \\ 1+\cos ^{2} x & \sin ^{2} x & \sin 2 x \\ \cos ^{2} x & \sin ^{2} x & 1+\sin 2 x\end{array}\right| \). Then the ordered pair \( (\mathrm{m}, \mathrm{M}) \) is equal to :
[JEE Main 2020, 6 Sep (Shift 1)]
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The function, \(f(x)=(3 x-7) x^{2 / 3}, x \in R\), is increasing for all \(x\) lying in:
[JEE Main 2020, 3 Sep (Shift 1)]
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The equation of the tangent to the curve \(y=b e^{-x / a}\) at the point where it crosses the \(y\)-axis is
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Let \(f:[-1,1] \rightarrow R\) be defined \(a\) s \(f(x)=a x^2+b x+c\) for all \(x \in[-1,1]\), where \(a, b, c \in \mathrm{R}\) such that \(f(-1)=2, f^{\prime}(-1)\) \(=1\) and for \(x \in(-1,1)\) the maximum value of \(f^{\prime \prime}(x)\) is \(\frac{1}{2}\). If \(f(x) \leq \alpha, x \in[-1,1]\), then the least value of \(\alpha\) is equal to........
[JEE Main 2021, 17 Mar (Shift 2)]
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Let \(f: R \rightarrow R\) be defined as
\[f(x)=\left\{\begin{aligned}-\frac{4}{3} x^3+2 x^2+3 x, & x>0 \\3 x e^x, & x \leq 0\end{aligned}\right.\]
Then \(f\) is increasing function in the interval.
[JEE Main 2021, 22 Jul (Shift 2)]
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Let \(g(x)=f(x)+f(1-x)\) and \(f^{\prime \prime}(x)>0, x \in(0,1)\). If \(\mathrm{g}\) is decreasing in the interval \((0, \alpha)\) and increasing in the interval \((\alpha, 1)\), then \(\tan ^{-1}(2 \alpha)+\tan ^{-1}\left(\frac{1}{\alpha}\right)+\tan ^{-1}\left(\frac{\alpha+1}{\alpha}\right)\) is equal to:
[JEE Main 2023, 10 Apr (Shift 2)]
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Let ' a ' be a real number such that the function \(f(x)=a{x}^{2}+6x-15,x\in R\) is increasing in \(\left(-\infty ,\frac{3}{4}\right)\) and decreasing in \(\left(\frac{3}{4},\infty \right)\). Then the function \(g(x)=a{x}^{2}-6x+15,x\in R\) has a :
[JEE Main 2021, 20 Jul (Shift 1)]
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Question. The function \(f(x)=3x+5\) in increasing for ..............
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Let the function \(f(x)=2 x^3+(2 p-7) x^2+3(2 p-9) x-6\) have a maxima for some value of \(x<0\) and a minima for some value of \(x>0\). Then, the set of all values of \(p\) is
[JEE Main 2023, 25 Jan (Shift 2)]
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Consider all rectangles lying in the region \(\left\{(\mathrm{x}, \mathrm{y}) \in \mathbb{R} \times \mathbb{R}: 0 \leq \mathrm{x} \leq \frac{\pi}{2} \text { and } 0 \leq \mathrm{y} \leq 2 \sin (2 \mathrm{x})\right\} \) and having one side on the \(\mathrm{x}\)-axis. The area of the rectangle which has the maximum perimeter among all such rectangles, is
[JEE Advanced 2020]
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Let the function \(f:(0, \pi) \rightarrow R\) be defined by \(f(\theta)=(\sin \theta+\cos \theta)^2+(\sin \theta-\cos \theta)^4\)
Suppose the function \(f\) has a local minimum at \(\theta\) precisely when \(\theta \in\left\{\lambda_1 \pi, \ldots . \lambda_{ r } \pi\right\}\), where \(0<\lambda_1<\ldots . .<\lambda_r<1\). Then the value of \(\lambda_1+\ldots .+\lambda_{ r }\) is
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The interval in which the function \(f\left(x\right)=\frac{4{x}^{2}+1}{x}\) is decreasing is:
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If the tangent to the curve, \(y=f(x)=x \log _e x,(x>0)\) at a point \((c, f(c))\) is parallel to the line-segment joining the points \((1,0)\) and \((e, e)\), then \(c\) is equal to:
[JEE Main 2020, 6 Sep (Shift 2)]
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Let \(f:(a, b) \rightarrow R\) be twice differentiable such that \(f(x)=\int_a^x g(t) d t\) for a differentiable function \(g(x)\). If \(f(x)=0\) has exactly five distinct roots in \((a, b)\), then \(g(x) g^{\prime}(x)=0\) has at least:
[JEE Main 2021, 27 Jul (Shift 2)]
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If the curve \( y=a x^{2}+b x+c, x \in R \), passes through the point \( (1,2) \) and the tangent line to this curve at origin is \( y=x \), then the possible values of \( a, b, c \) are:
[JEE Main 2021, 24 Feb (Shift 2)]
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A wire of length \(20 m\) is to be cut into two pieces. A piece of length \(l_1\) is bent to make a square of area \(A_1\) and the other piece of length \(l_2\), is made into a circle of area \(A_2\). If \(2 A_1+3 A_2\) is minimum then \(\left(\pi l_1\right): l_2\) is equal to:
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The range of \(a\in \mathrm{ℝ}\) for which the function \(f(x)=(4a-3)(x+{\log }_{e}5)+2(a-7)cot\left(\frac{x}{2}\right){\sin }^{2}\left(\frac{x}{2}\right)\) \(x\neq 2n\pi ,\mathrm{n}\in \mathrm{ℕ}\), has critical points, is:
[JEE Main 2021, 16 Mar (Shift 1)]
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If the tangent at \(\ P(1,1) \) on \(\ y^2=x(2-x)^2 \) meets the curve again at \(\ Q \), then \(\ Q \) is
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If a curve passes through the origin and the slope of the tangent to it at any point \((x, y)\) is \(\frac{x^2-4 x+y+8}{x-2}\), then this curve also passes through the point:
[JEE Main 2021, 25 Feb (Shift 1)]
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\(\ \mathbf{f}(\mathbf{x})=\left(\frac{\mathrm{e}^{2 x}-1}{\mathrm{e}^{2 x}+1}\right) \) is
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An angle of intersection of the curves, \(\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1\) and \({x}^{2}+{y}^{2}=ab,a>b\) is:
[JEE Main 2021, 31 Aug (Shift 2)]
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The number of points on the curve \(y=54 x^5-135 x^4-70 x^3\) \(+180 x^2+210 x\) at which the normal lines are parallel to \(x+90 y+2=0\) is:
[JEE Main 2023, 30 Jan (Shift 1)]
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Suppose \( f(x) \) is a polynomial of degree four, having critical points at \( -1,0,1 \). If \( \mathrm{T}=\{\mathrm{x} \in \mathrm{R} \mid \mathrm{f}(\mathrm{x})=\mathrm{f}(0)\} \), then the sum of squares of all the elements of \( T \) is :
[JEE Main 2020, 3 Sep (Shift 2)]
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Consider the function \(f: R \rightarrow R\) defined by \(f(x)=\left\{\begin{array}{ll}\left(2-\sin \left(\frac{1}{x}\right)\right)|x|, & x \neq 0 \\ 0, & x=0\end{array}\right.\). Then \(f\) is:
[JEE Main 2021, 17 Mar (Shift 2)]
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For which of the following curves, the line \(x+\sqrt{3 y}=2 \sqrt{3}\) is the tangent at the point \(\left(\frac{3 \sqrt{3}}{2}, \frac{1}{2}\right)\) ?
[JEE Main 2021, 24 Feb (Shift 2)]
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The points at which the tangent passes through the origin for the curve \(\ y=4 x^3-2 x^5 \) are
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If Rolle’s theorem holds for the function \(f(x) = \ x^{3} - ax^{2} + bx - 4,\) \(x\epsilon\lbrack 1,2\rbrack\) with \(f'(\frac{4}{3}) = 0,\ \)then ordered pair (a, b) is equal to
[JEE Main 2021, 25 Feb (Shift 1)]
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Let \(A=\left[a_{i j}\right]\) be a \(3 \times 3\) matrix, where
\(a_{i j}=\left\{\begin{array}{cc}1, & \text { if } i=j \\-x, & \text { if }|i-j|=1 \\2 x+1, & \text { otherwise }\end{array}\right.\)
Let a function \(f: R \rightarrow R\) be defined as \(f(x)=\operatorname{det}(A)\). Then the sum of maximum and minimum values of \(f\) on \(R\) is equal to:
[JEE Main 2021, 20 Jul (Shift 1)]
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Let \(f:[-1,1] \rightarrow R\) be defined as \(f(x)=a x^2+b x+c\) for all \(x \in[-1,1]\), where \(a, b, c \in \mathrm{R}\) such that \(f(-1)=2, f^{\prime}(-1)\) \(=1\) and for \(x \in(-1,1)\) the maximum value of \(f^{\prime \prime}(x)\) is \(\frac{1}{2}\). If \(f(x) \leq \alpha, x \in[-1,1]\), then the least value of \(\alpha\) is equal to........
[JEE Main 2021, 17 Mar (Shift 2)]
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The sum of all the local minimum values of the twice differentiable function \(f: R \rightarrow R\) defined by \(f(x)=x^3-3 x^2-\frac{3 f^{\prime \prime}(2)}{2} x+f^{\prime \prime}(1)\) is :
[JEE Main 2021, 20 Jul (Shift 2)]
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If the radius of a sphere is measured as 9 cm with an error of 0.03 cm , then find the approximating error in calculating its volume.
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If the surface area of a cube is increasing at a rate of \( 3.6 \mathrm{~cm}^{2} / \mathrm{sec} \), retaining its shape; then the rate of change of its volume (in \( \mathrm{cm}^{3} / \mathrm{sec} \) ), when the length of a side of the cube is \( 10 \mathrm{~cm} \), is:
[JEE Main 2020, 3 Sep (Shift 2)]
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Let \( f:(-1, \infty) \rightarrow R \) be defined by \( f(0)=1 \) and \( f(x)= \)\( \frac{1}{x} \log _{e}(1+x), x \neq 0 \). Then the function \( f \) :
[JEE Main 2020, 2 Sep (Shift 2)]
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The sum of the abosolute maximum and minimum values of the function \(f(x)=\left|x^2-5 x+6\right|-3 x+2\) in the interval \([-1,3]\) is equal to:
[JEE Main 2023, 1 Feb (Shift 2)]
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Let \( f(x)=x \cos ^{-1}(-\sin |x|), x \in\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \), then which of the following is true
[JEE Main 2020, 8 Jan (Shift 1)]
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The area (in sq. units) of the largest rectangle \( A B C D \) whose vertices \( A \) and \( B \) lie on the \( x \)-axis and vertices \( C \) and \( D \) lie on the parabola, \( y=x^{2}-1 \) below the \( x \)-axis is
[JEE Main 2020, 4 Sep (Shift 2)]
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Let the function \(f(x)=2 x^3+(2 p-7) x^2+3(2 p-9) x-6\) have a maxima for some value of \(x<0\) and a minima for some value of \(x>0\). Then, the set of all values of \(p\) is
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The local maximum value of the function \(f(x)=\left(\frac{2}{x}\right)^{x^2}, x>0\), is:
[JEE Main 2021, 26 Aug (Shift 2)]
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If \(x=1\) is a critical point of the function \(f(x)=\left(3 x^2+a x-2\right.\) -a) \(e^x\), then:
[JEE Main 2020, 5 Sep (Shift 2)]
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The function \(f(x)=\tan ^{-1}(\sin x+\cos x)\) is an increasing function in
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If the tangent to the curve \(y = x^{3}\) at the point \(P\left( t,t^{3} \right)\) meets the curve again at \(Q\), then the ordinate of the point which divides \(\text{PQ}\) internally in the ratio \(1:2\) is :
[JEE Main 2021, 24 Feb (Shift 1)]
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If a curve passes through the origin and the slope of the tangent to it at any point \((x, y)\) is \(\frac{x^2-4 x+y+8}{x-2}\), then this curve also passes through the point:
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Let \(f(x)=3\left({\cos }^{4}x−2{\cos }^{2}x+1\right)+10{\sin }^{3}x+6{\sin }^{2}x−3,\) \(x\in \left[−\frac{x}{6},\frac{x}{2}\right]\). Then, \(f\) is:
[JEE Main 2021, 25 Jul (Shift 1)]
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Let quadratic curve passing through the point \((-1,0)\) and touching the line \(y=x\) at \((1,1)\) be \(\mathrm{y}=f(x)\). Then the \(x\)-intercept of the normal to the curve at the point \((\alpha, \alpha+1)\) in the first quadrant is.........
[JEE Main 2023, 10 Apr (Shift 2)]
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\(f(x)=\left(\frac{{e}^{2x}−1}{{e}^{2x}+1}\right)\) is
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Let slope of the tangent line to a curve at any point \( P(x, y) \) be given by \( \frac{x y^{2}+y}{x} \). If the curve intersects the line \( x+2 y=4 \) at \( x=-2 \), then the value of \( y \), for which the point \( (3, y) \) lies on the curve, is:
[JEE Main 2021, 26 Feb (Shift 2)]
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Let f and g be twice differentiable functions on R such that
\(f"(x)=g"(x)+6x\)
\(f'(1)=4g'(1)−3=9\)
\(f(2)=3g(2)=12\)
Then which of the following is NOT true?
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Consider the function \(f: R \rightarrow R\) defined by \(f(x)=\left\{\begin{array}{ll}\left(2-\sin \left(\frac{1}{x}\right)\right)|x|, & x \neq 0 \\ 0, & x=0\end{array}\right.\).Then \(f\) is:
[JEE Main 2021, 17 Mar (Shift 2)]
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Let \(f: R \rightarrow R\) be defined as \(f(x)=\left\{\begin{aligned}-\frac{4}{3} x^3+2 x^2+3 x, & x>0 \\3 x e^x, & x \leq 0\end{aligned}\right.\)
Then \(f\) is increasing function in the interval.
[JEE Main 2021, 22 Jul (Shift 2)]
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If the functions \(f(x)=\frac{x^3}{3}+2 b x+\frac{a x^2}{2}\) and \(g(x)=\frac{x^3}{3}+a x+b x^2, a \neq 2 b\) have a common extreme point, then \(a+2 b+7\) is equal to
[JEE Main 2023, 30 Jan (Shift 2)]
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If the local maximum value of the function \(f(x)=\left(\frac{\sqrt{3 e}}{2 \sin x}\right)^{\sin ^2 x}, x \in\left(0, \frac{\pi}{2}\right)\) is \(\frac{k}{e}\), then \(\left(\frac{k}{e}\right)^8+\frac{k^8}{e^5}+k^8\) is equal to
[JEE Main 2023, 12 Apr (Shift 1)]
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If the curve \(y=a x^2+b x+c, x \in R\) passes through the point \((1,2)\) and the tangent line to this curve at origin is \(y=x\), then the possible values of \(a, b, c\) are:
[JEE Main 2021, 24 Feb (Shift 2)]
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Let \( P(h, k) \) be a point on the curve \( y=x^{2}+7 x+2 \), nearest to the line, \( y=3 x-3 \). Then the equation of the normal to the curve at \( \mathrm{P} \) is:
[JEE Main 2020, 2 Sep (Shift 1)]
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If the tangent to the curve \( y=x+\sin y \) at a point \( (a, b) \) is parallel to the line joining \( \left(0, \frac{3}{2}\right) \) and \( \left(\frac{1}{2}, 2\right) \), then :
[JEE Main 2020, 2 Sep (Shift 1)]
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A spherical iron ball \( 10 \mathrm{~cm} \) in radius is coated with a layer of ice of uniform thickness that melts at a rate of \( 50 \mathrm{~cm}^{3} / \mathrm{min} \). When the thickness of ice is \( 5 \mathrm{~cm} \), then the rate at which the thickness of ice decreases, is
[JEE Main 2020, 9 Jan (Shift 1)]
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Let \( a \) be a real number such that the function \( f(x)=a x^{2}+6 x-15, x \in R \) is increasing in \( \left(-\infty, \frac{3}{4}\right) \) and decreasing in \( \left(\frac{3}{4}, \infty\right) \). Then the function \( g(x)=a x^{2}-6 x+15, x \in R \) has a:
[JEE Main 2021, 20 Jul (Shift 1)]
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The position of a moving car at time \( t \) is given by \( f(t)=a t^{2}+b t+c, t>0 \), where \( a \), \( b \) and \( c \) are real numbers greater than 1. Then the average speed of the car over the time interval \( \left[t_{1}, t_{2}\right] \) is attained at the point :
[JEE Main 2020, 6 Sep (Shift 1)]
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If the equation of the normal to the curve \(y=\frac{x-a}{(x+b)(x-2)}\) at the point \((1,-3)\) is \(x-4 y=13\), then the value of \(a+b\) is equal to........
[JEE Main 2023, 29 Jan (Shift 2)]
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Which of the following points lies on the tangent to the curve \( x^{4} e^{y}+2 \sqrt{y+1}=3 \) at the point \( (1,0) \) ?
[JEE Main 2020, 5 Sep (Shift 2)]
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Let the function \( f:[-7,0] \rightarrow R \) be continuous on \( [-7 \), 0 ] and differentiable on \( (-7,0) \). If \( f(-7)=-3 \) and \( f^{\prime}(x) \leq \) 2 for all \( x \in(-7,0) \), then for all such functions \( f, f(-1)+f(0) \) lies in the interval
[JEE Main 2020, 7 Jan (Shift 1)]
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The function \(f(x)=x^3-6 x^2+a x+b\) is such that \(f(2)=f(4)=0\). [JEE Main 2021, 1 Sep (Shift 2)]
Assertion (A): there exists \(x_1, x_2 \in(2,4), x_1
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The value of \(a\in R\) for which the function \(\displaystyle f\left ( x \right )= \left ( 4a-3 \right )\left ( x+\log 5 \right )+2\left ( a-7 \right )\cot\left ( \frac{x}{2} \right )\sin ^{2}\left ( \frac{x}{2} \right )\) ,\(x\neq 2n\pi ,n\in N\) has critical points is
[JEE Main 2021, 16 Mar (Shift 1)]
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Let \(f:(0,1) \rightarrow \mathrm{R}\) be a function defined by \(f(x)=\frac{1}{1-e^{-x}}\) and \(g(x)=(f(-x)-f(x))\).
Assertion (A): \(g\) is an increasing function in \((0,1)\)
Reason (R): \(g\) is one-one in \((0,1)\)
Then,
[JEE Main 2023, 25 Jan (Shift 1)]
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The set of all real values of \( \lambda \) for which the function \( f(x) \)\( =\left(1-\cos ^{2} x\right) \cdot(\lambda+\sin x), x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \) has exactly one maxima and exactly one minima, is :
[JEE Main 2020, 6 Sep (Shift 2)]
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