Question

If $0<a<1$ and $f$ is defined as $f\left(x\right)=\{\begin{array}{c}{x}^a;ifx>0\\ 0;ifx=0\end{array}$

then?

• A. $f$ is differentiable at $x=0$
• B. $f$ is not continuous
• C. $f$ is not differentiable at $x=0$
• D. $f$ is bounded

Answer 1

The correct option is C $f$ is not differentiable at $x=0$

$Forthegivenfunctionletuscheckthelefthandlimitandtherighthandlimit,whetherthevaluewouldbeequalIftheL.H.L=R.H.LThenitshouldbedifferentiableatthegivenpointLefthandlimit\underset{a⟶0}{lim}{x}^a=x^0=1Righthandlimit\underset{a⟶0}{lim}0=0SinceL.H.L\ne R.H.LThefunctionisnotdifferentiableat0<a<1$