Question

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. $h'(x) \neq 1$, for all $x>1$
• D. $h$ is not differentiable at two values of $x$

Answer 1

Answer: (D)

$h(x)=\min \left(x, x^{2}\right)=\begin{cases}x, & x<0 \\ x^{2}, & 0 \leq x<1 \\ x, & x \geq 1\end{cases}$

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 $h'(x)=1, \forall x \geq 1$