maximum and minimun of modular function $f(x) = |(x - 2)^2 - 8|$

Question

Given the follow function: $ f(x) = | (x-2)^2 - 8 | $

What are the critical points? Classify they as relative maximum, relative minimum, global maximum and global minimum.

For last describe if the function is convex.

I know that is a simple problem, but I'm not secure about to work with modular functions, I want understand the full nature of modular derivative.

Answer 1

The function touches the $x$-axis where $(x-2)^2=8$ $\implies$ $x=2\pm2\sqrt2$. These points $(2\pm2\sqrt2,0)$ are obviously global minima as $f(x)\ge0$.

For $x<2-2\sqrt2$ and $x>2+2\sqrt2$, $f(x)=(x-2)^2-8$ has no critical points.

For $2-2\sqrt2< x < 2+2\sqrt2$, $f(x)=8-(x-2)^2$ has a local maximum $(2,8)$ at $x=2$.