Let denote the set of all real numbers. Define the function f: R R by f(x)= array cc 2-2…

Mathematics · JEE Advanced · NTA Exams — Limit, Continuity and Differentiability

Let denote the set of all real numbers. Define the function \(f: \mathbb{R} \rightarrow \mathbb{R}\) by
\[f(x)=\left\{\begin{array}{cc} 2-2 x^{2}-x^{2} \sin \frac{1}{x} & \ ext { if } x \ eq 0 \\ 2 & \ ext { if } x=0 \end{array}\right.\]
Then which one of the following statements is TRUE?

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