Question

Let $f\left(x\right)=\{\begin{array}{cc}{x}^2\left|\cos \frac{\pi }{x}\right|,& x\ne 0\\ 0,& x=0\end{array},x\in R$ then $f$ is

• A. Differentiable at $x=0$ but not differentiable at $x=2$
• B. Differentiable both at $x=0$ and at $x=2$
• C. Not differentiable at $x=0$ but differentiable at $x=2$
• D. Differentiable neither at $x=0$ nor at $x=2$

Answer 1

The correct option is A Differentiable at $x=0$ but not differentiable at $x=2$

$f\left(x\right)=\{\begin{array}{cc}{x}^2\left|\cos \frac{\pi }{x}\right|,& x\ne 0\\ 0,& x=0\end{array},x\in R$
Let $p\left(x\right)=x^2$ and $g\left(x\right)=\left|\cos \frac{\pi }{x}\right|$
Now $g\left(x\right)$ is a continuous non-differentiable at $x=0$, and $2$ but since $p\left(0\right)=0$, so, $p\left(x\right).g\left(x\right)$ is non-differentiable at $x=2$.