Question

Let $\mathrm{h}\left(\mathrm{x}\right)=\min \left\{\mathrm{x},{\mathrm{x}}^2\right\}$, for every real number $\mathrm{x}$. Then:

• A. $\mathrm{h}$ is discontinuous for all $\mathrm{x}$
• B. $\mathrm{h}$ is differentiable for all $\mathrm{x}$
• C. $\mathrm{h}’\left(\mathrm{x}\right)=2,\mathrm{for}\mathrm{all}\mathrm{x}>1$
• D. $\mathrm{h}$ is not differentiable at two values of $\mathrm{x}$

Answer 1

The correct option is D

$\mathrm{h}$ is not differentiable at two values of $\mathrm{x}$

Determine the relation between $h\text{and}x$.

We have, $\mathrm{h}\left(\mathrm{x}\right)=\min (\mathrm{x},{\mathrm{x}}^2)$. We can write this function in different forms as shown below:

$\mathrm{h}\left(\mathrm{x}\right)=\left\{\begin{array}{l}\mathrm{x}\mathrm{x}<0\\ {\mathrm{x}}^{2}0\le \mathrm{x}<1\\ \mathrm{x}\mathrm{x}\ge 1\end{array}\right.$

The graph for the given function is as shown:

As we can see from the graph, the function is not differentiable at two points which are $\mathrm{x}=0$ and $\mathrm{x}=1$, Also, at $x\ge 1$, the function represents a straight line having a slope $1$, therefore $h\prime \left(x\right)=1,\forall x\ge 1$

Hence, option D is the correct answer.