Question

Let $f:R\to R$ be a function defined
as $f(x)=\{\begin{array}{ll}3\left(1-\frac{|x|}{2}\right)& \text{ if }|x|\le 2\\ 0& \text{ if }|x|>2\end{array}$
Let $g:R\to R$ be given by $g(x)=f(x+2)-f(x-2)$.
If $n$ and $m$ denote the number of points in $R$ where $g$ is not continuous and not differentiable, respectively, then $n+m$ is equal to _______.

Answer 1

Answer: 4

$f(x-2)\{\begin{array}{ll}\frac{3x}{2}& -4\le x\le -2\\ -\frac{3x}{2}& -2<x\le 0\\ 0& x\in (-\infty ,-4)\cup (0,+\infty )\end{array}$
$f(x-2)\{\begin{array}{ll}\frac{3x}{2}& 0\le x\le 2\\ -\frac{3x}{2}+6& 2\le x\le 4\\ 0& x\in (-\infty ,0)\cup (4,+\infty )\end{array}$
$g(x)=f(x+2)-f(x-2)\{\begin{array}{ll}3x2+6& -4\le x\le -2\\ -\frac{3x}{2}& -2<x<2\\ \frac{3x}{2}-6& 2\le x\le 4\\ 0& x\in (-\infty ,-4)\cup (4,\infty )\end{array}$

$n=0$
$m=4\Rightarrow (n+m=4)$