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.
So, plotting curves for $-|x|,-\sqrt{1-x^2}$ and representing $max\{-|x|,-\sqrt{1-x^2}\}$ through solid line we get curve as

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