Question

lf $f\left(x\right)=\{\begin{array}{ccc}{x}^{n}cos\frac{1}{x}& if& x\ne 0\\ 0& if& x=0\end{array}$and $n>0$,
then $f\left(x\right)$ is differentiable at $x=0$ if

• A. $n\in (0,\infty )$
• B. $n\in (1,3]$
• C. $n=1$
• D. $n\in (2,\infty )$

Answer 1

The correct option is D $n\in (2,\infty )$

$f^{\prime }\left(x\right)=n{x}^{n-1}\cos \left(\frac{1}{x}\right)+\frac{x^n}{x^2}\sin \left(\frac{1}{x}\right)$

$=n{x}^{n-1}cos\left(\frac{1}{x}\right)+x^{n-2}\sin \left(\frac{1}{x}\right)$

It will be differentiable only If Lim$f^{\prime }\left(x\right)=0$

$x\to 0$

So coefficient of $x$ should be greater then $0$ otherwise above limit will not exist.

$n-1>0andn-2>0$

$n-2>0$

$n>2$