Question

Given $f\left(x\right)$ is a function such that $f\left(x\right)=$ $[\begin{array}{c}{x}^{\alpha }\sin \frac{1}{x}ifx>0\\ 0ifx=0\end{array}$ where $\alpha$ is a constant. If $f\left(x\right)$ is a derivable $\forall x\ge 0,$ then the least integral value of $\alpha$ is

• A. $1$
• B. $2$
• C. $0$
• D. none of these

Answer 1

Answer: (A)

$1$

Since f is derivable thus f is continuous at $0$.

$\underset{x\to 0}{lim}f\left(x\right)$

$=\underset{x\to 0}{lim}{x}^{\alpha }\sin \frac{1}{x}=0$ if $\alpha >0$

$\Rightarrow \alpha \ge 1$

$\Rightarrow$ Minimum value of $\alpha =1$.