Square root of the max is the max of the square root? I apologize if the question seems so obvious, but I don't have a strong base on maths nor I know the tools to prove this simple statement.
For a given function $f(x)$, is it true that
$$ \left(\max |f(x)|^2 \right)^{\frac{1}{2}} = \max |f(x)|.$$
I wonder if it should have a less or equal sign instead, but I do not know where to start.
Thanks in advance. 
Edit: I see that I was misunderstanding the answers... I thought my claim was false and should be $\leq$ instead. Now I see that equality holds by proving the inequalities in both ways. Thanks a lot!

Short answer:
$$|a|>|b|\iff a^2>b^2.$$
