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

• A

  $h$ is not continuous for all $x$
• B

  $h$ is differentiable for all $x$
• C

  $h^{\prime}(x) \neq 1$, for all $x\gt1$
• Correct answer
  D

  $h$ is not differentiable at two values of $x$

Solution

$h(x)=\min \left(x, x^2\right)=\left\{\begin{array}{cl}x, & x \lt 0 \\ x^2, & 0 \leq x \lt 1 \\ x, & x \geq 1\end{array}\right.$
It is evident from the above 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 $x \geq 1$, the function represents a straight line having slope 1 , therefore $h^{\prime}(x)=1, \forall x \geq 1$.

Hence, the correct answer is option: 'D'.