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EEJ MAIN Mathematics QUESTION #6031
Question 1

Let $f(x)=\begin{cases}\max\{|x|,x^2\},&|x|\le2\\8-2|x|,&2<|x|\le4\end{cases}$. Let $S$ be the set of points in $(-4,4)$ where $f$ is not differentiable. Find $S$.

  • Empty set
  • $\{-2,-1,0,1,2\}$✔️
  • $\{-2,-1,1,2\}$
  • $\{-2,2\}$
Correct Answer Explanation

On $[-2,2]$: $\max(|x|,x^2)=x^2$ when $x^2\ge|x|$, i.e., $|x|\ge1$; equals $|x|$ when $|x|\le1$.

So $f(x)=|x|$ for $|x|\le1$ and $f(x)=x^2$ for $1\le|x|\le2$.

Check non-differentiability: At $x=0$: $|x|$ is not differentiable. At $x=\pm1$: left-hand $f'=\pm1$, right-hand $f'=2x=\pm2$ — not equal, so not differentiable. At $x=\pm2$: $f(2^-)=4$, $f(2^+)=4$; $f'(2^-)=4$, $f'(2^+)=-2$ — not differentiable.

So $S=\{-2,-1,0,1,2\}$.