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. $h^{\prime }(x)\ne 1$, for all $x>1$
• D. $h$ is not differentiable at two values of $x$

Answer 1

Answer: (D)

$h(x)=\min \left(x,x^2\right)=\{\begin{array}{ll}x,& x<0\\ x^2,& 0\le x<1\\ x,& x\ge 1\end{array}$

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 $x\ge 1$, the function represents a straight line having slope $1$ ,
therefore $h^{\prime }(x)=1,\forall x\ge 1$