Question

Let $h\left(x\right)=min(x,x^2)$, for every real numbers $x$. Then

• A. $h$ is not continuous for all $x$
• B. $h$ is differentiable for all $x$
• C. $h^{\prime }\left(x\right)\ne 1$, for all $x>1$
• D. $h$ is not differentialble at two values of $x$

Answer 1

The correct option is D $h$ is not differentialble at two values of $x$

$h\left(x\right)=min(x,x^2)=h\{\begin{array}{c}x,x<0\\ x^2,0\le x<1\\ x,x\ge 1\end{array}$
It is evident from the graph that the given function is continuous for all $x$.
Since, there are sharp edges at $x=0$ and $x=1$, the function is not differentiable at these points
Also, at $x\ge 1$, the function represents a straight line having slope $1$, therefore $h^{\prime }\left(x\right)=1,\forall x\ge 1$