Question

Let $f:(-1,1)\to R$ be a function defined by $f\left(x\right)=max\{-|x|,-\sqrt{1-x^2}\}$. If $K$ be the set of all points at which $f$ is not differentiable, then $K$ has exactly

• A. one element
• B. two elements
• C. three elements
• D. five elements

Answer 1

The correct option is C three elements

Given
$f:(-1,1)\to R$
$f\left(x\right)=max\{-|x|,-\sqrt{1-x^2}\}$

$y=-\left|x\right|=\{\begin{array}{cc}-x,& \text{if}x\ge 0\\ x,& \text{if}x<0\end{array}$

and $y=-\sqrt{1-x^2}\Rightarrow y^2+x^2=1$
The above equation represents a semicircle with unit radius below the $x$-axis.
Therefore, we have the graph -

Clearly, from the graph, $f\left(x\right)$ is not differentiable at three points $A,O,B.$