Let \mathbb{R} denote the set of all real numbers. Define the function f: \mathbb{R} \rightarrow \mathbb{R} by f(x)={\\begin{array}{c}2-2 x^2-x^2 \sin \\\\\\{1\end\frac{array}{x} , \text{ if } x \neq 0 ; 2 , \text{ if } x=0 } Then which one of the following statements is TRUE? [JEE Adv 2025 P1]

Correct Answer is : There is a positive real number δ, such that f is a decreasing function on the interval (0,δ)

The function f is NOT differentiable at x=0

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

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

x=0 is a point of local minima of f

Limits, Continuity and Differentiability

Section: Mathematics

Question : 2 of 51

Marks: +1, -0

Let ℝ denote the set of all real numbers. Define the function f:ℝ⟶ℝ by f(x)={

2−2x2−x2sin‌‌

‌ if ‌x≠0

‌ if ‌x=0

Then which one of the following statements is TRUE?

[JEE Adv 2025 P1]

The function f is NOT differentiable at x=0
There is a positive real number δ, such that f is a decreasing function on the interval (0,δ)
For any positive real number δ, the function f is NOT an increasing function on the interval (−δ,0)
x=0 is a point of local minima of f

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