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

(A)
RHD at \(x=0: \lim _{h \rightarrow 0} \frac{\left(2-2 h^2-h^2 \sin \frac{1}{h}\right)-2}{h}=0\)
Similarly LHD at \(\mathrm{x}=0\) is also equal to 0 .
\(\therefore\) Differentiable at \(\mathrm{x}=0\)
(B) \(f^{\prime}(x)=-4 x-2 x \sin \frac{1}{x}-x^2\left(\cos \frac{1}{x}\right)\left(\frac{-1}{x^2}\right)\)
\(\begin{aligned}
& f^{\prime}(x)=-\left(4 x+2 x \sin \frac{1}{x}\right)+\cos \frac{1}{x} \\
& f^{\prime}(x)=-\left(2 x\left(4-\sin \frac{1}{x}\right)\right)+\cos \frac{1}{x}
\end{aligned}\)
for \(\mathrm{x} \in(0, \delta)\)
\(\Rightarrow\) We can't say \(f(x)\) is decreasing on \((0, \delta)\) as \(\cos \frac{1}{x}\) oscillates.
(C) for \(x \in(-\delta, 0)\), for any \(\delta>0\)
\(\Rightarrow f(x)\) is not increasing on \((-\delta, 0)\) as \(\cos \frac{1}{x}\) oscillates from -1 to 1.
(D) \(f(0)=2\)
\(\begin{aligned}
& f(0+h) <2 \\
& f(0-h) <2
\end{aligned}\)
\(\therefore \mathrm{x}=0\) is local maxima