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Limits Continuity and Differentiability question

2025 · Shift 1 · Q19

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PYQ Library
JEE Advanced
Mathematics
Limits Continuity and Differentiability
2025 · Shift 1 · Q19

Limits Continuity and Differentiability question

2025 · Shift 1 · Q19

JEE AdvancedMathematicsLimits Continuity and DifferentiabilityMCQ+3 / −1

Let

\mathbb{R}

denote the set of all real numbers. Define the function

f : \mathbb{R} \to \mathbb{R}

by

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

Then which one of the following statements is TRUE?

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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