If $x_0$ minimizes $f^2$, $x_0$ minimizes $f$.

Question

I feel that this statement is true, but I am not convinced. Is there a simple counterexample or how might this statement be formalized to make it true?

Answer 1

There are a lot of counter examples..

Take $f=\sin(x)$, it is obvious the statement does not hold. Since $f^2$ is positive, it is minimized when $\sin(x)=0$, but minimal value of $\sin(x)$ is actually $-1$. So the values that minimize $\sin^2(x)$ can never minimize $\sin(x)$.

Answer 2

The statement is generally not true.

Minimizing $f^2$ is equivalent to minimizing $|f|$ since $g(x)=x^2$ is an increasing function over the nonnegative domain.

However, this is not true if $f$ can take negative value as $g(x)=x^2$ is no longer an increasing function over the whole $\mathbb{R}$.