Question

Let $y=min\{x,x^2,x^3\}$. At how many points in the interval $(-1,1]$, $y$ is not differentiable?

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is B $2$

$y=min\{x,x^2,x^3\}$
Given function can be written as
$y=\{\begin{array}{cc}{x}^3;& x<-1\\ x;& -1\le x<0\\ x^3;& 0\le x<1\\ x;& x\ge 1\end{array}$
This function is continuous $\forall x\in R$
$y^{\prime }=\{\begin{array}{cc}3x^2;& x<-1\\ 1;& -1\le x<0\\ 3x^2;& 0\le x<1\\ 1;& x\ge 1\end{array}$
Therefore, function is not differentiable at a total of three points $x=-1,0,1$ of which two point $x=0,1\in (-1,1]$