Question

is derivable and has a continuous derivative at $x=0$

• A. $m$ $\in (1,\infty )$
• B. $m$ $\in [2,\infty )$
• C. $m$ $\in (2,\infty )$
• D. $m$ $\in (-\infty ,2)$

Answer 1

The correct option is C $m$ $\in (2,\infty )$

For the function to be derivable at $x=0,f^{\prime }\left(0^{+}\right)$ i.e. $\frac{h^m\sin \frac{1}{h}}{h}$ has to be $0$ as $h\to 0.$
Since $\sin \frac{1}{h}$ takes any value from $[-1,1]$, so $m\ge 2.$
Now, $f^{\prime }\left(x\right)=-h^{(m-2)}\times \cos \frac{1}{h}+m{h}^{(m-1)}\times \sin \frac{1}{h}$.
For this to be continuous at $x=0,m-2>0$ implying $m\in (2,\infty )$.