Question

Let $f:R\to R$ be a function defined by $f\left(x\right)=max\left\{x,x^3\right\}.$ The set of all the points where $f\left(x\right)$ is not differentiable is:

• A. $\left\{-1,1\right\}$
• B. $\left\{-1,0\right\}$
• C. $\left\{0,1\right\}$
• D. $\left\{-1,0,1\right\}$

Answer 1

The correct option is D

$\left\{-1,0,1\right\}$

Explanation for the correct option:

Determine the set of all the points where $f\left(x\right)$ is not differentiable.

We have, $f\left(x\right)=max\left\{x,x^3\right\}$, that can be written as:

$\therefore f\left(x\right)=\left\{\begin{array}{l}x,x\le -1\\ x^3,-1\le x\le 0\\ x,0\le x\le 1\\ x^3,x\ge 1\end{array}\right.$

Differentiating $f\left(x\right)$, we have,

$\therefore f\left(x\right)=\left\{\begin{array}{l}1,x\le -1\\ 3x^2,-1\le x\le 0\\ 1,0\le x\le 1\\ 3x^2,x\ge 1\end{array}\right.$

A function is not differentiable at points where the continuity breaks or where the graph has pointed ends and there is slope change.

Therefore, we can say that the function is not differentiable at points where the definition of the function changes. We can see from the above equations that the function is not differentiable at $-1,0,1$ as the function is defined by different equations at these points.

Hence, the correct answer is Option (D).