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🛠️ JEE➗ Maths

Let \(\mathbb{R}\) denote the set of all real numbers. Define the function \(f:\mathrm{ℝ}\to \mathrm{ℝ}\) by \(f\left(x\…

Q1

Let \(\mathbb{R}\) denote the set of all real numbers. Define the function \(f:\mathrm{ℝ}\to \mathrm{ℝ}\) by

\(f\left(x\right)=\left{\begin{matrix}2-2{x}^{2}-{x}^{2}\sin \frac{1}{x}, & \text{ if }x\neq 0 \\ 2, & \text{ if }x=0\end{matrix}\right.\)

Then which one of the following statements is TRUE?

[JEE Advanced 2025]

a

The function \(f\) is NOT differentiable at \(x=0\)

b

There is a positive real number \(\delta\), such that \(f\) is a decreasing function on the interval \((0, \delta)\)

c

For any positive real number \(\delta\), the function \(f\) is NOT an increasing function on the interval \((-\delta ,0)\)

d

\(x=0\) is a point of local minima of \(f\)

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