Question

Let $f:R\to R$ and $g:R\to R$ be respectively given by $f\left(x\right)=|x|+1$ and $g\left(x\right)=x^2+1$. Define $h:R\to R$ by
$h\left(x\right)=\{\begin{array}{c}max\left\{f\right(x),g(x\left)\right\},x\le 0\\ min\left\{f\right(x),g(x\left)\right\},x>0.\end{array}$
The number of points at which $h\left(x\right)$ is not differentiable is

Answer 1

$f\left(x\right)=|x|+1$
$\Rightarrow f\left(x\right)=\{\begin{array}{c}1-x,x\le 0\\ x+1,x>0\end{array}$
$g\left(x\right)=x^2+1$
$h\left(x\right)=\{\begin{array}{c}max\left\{f\right(x),g(x\left)\right\},x\le 0\\ min\left\{f\right(x),g(x\left)\right\},x>0\end{array}$

The graph of $h\left(x\right)$ is represented through solid lines in the below curve,




Hence, there are $3$ points of non-differentiability for $h\left(x\right)$ at $x=\pm 1,0$.