Let $f(x)$ be a continuously differentiable function on the interval $(0,\infty )$ such that $f(1)=2$ and $\lim _{t\to x}\frac{{t}^{10}f(x)-{x}^{10}f(t)}{{t}^{9}-{x}^{9}}=1$ for each $x0.$ Then, for all $x0, f(x)$ is equal to [JEE Advanced 2024]

Question

Let $f(x)$ be a continuously differentiable function on the interval $(0,\infty )$ such that $f(1)=2$ and $\lim _{t\to x}\frac{{t}^{10}f(x)-{x}^{10}f(t)}{{t}^{9}-{x}^{9}}=1$ for each $x>0.$ Then, for all $x>0, f(x)$ is equal to [JEE Advanced 2024]

YESPYQ · Accepted Answer

Correct answer: b) $\frac{9}{11x}+\frac{13}{11}{x}^{10}$. $\lim_{t\to x}\frac{10t^9f(x)-f'(t)x^{10}}{9t^8}=1$ $\Rightarrow 10x^9f(x)-f'(x)x^{10}=9x^8$ $\Rightarrow f'(x)-\frac{10}{x}f(x)=-\frac{9}{x^2}$ Let $y=f(x)$. $IF=e^{-\int \frac{10}{x}\,dx}$ $IF=\frac{1}{x^{10}}$ $\frac{y}{x^{10}}=\int -\frac{9}{x^2}\cdot \frac{1}{x^{10}}\,dx$ $\frac{y}{x^{10}}=-9\int x^{-12}\,dx$ $\frac{y}{x^{10}}=\frac{9}{11}x^{-11}+C$ $\because y(1)=2\Rightarrow C=\frac{13}{11}$ $\Rightarrow y=\frac{9}{11x}+\frac{13}{11}x^{10}$

Let \(f(x)\) be a continuously differentiable function on the interval \((0,\infty )\) such that \(f(1)=2\) and \(\lim _…

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