Max of difference and difference of max

Question

Considerva function $f(x)$ and a function g(x) and suppose they both have a maximum. They may take positive or negative values.

Is it true that $$ \max_x (g(x)-f(x))\leq max_x g(x)-\max_x f(x) $$ Or the way around holds? Or we can't say?

Answer 1

Let $f(x) = 1-x^2$ throughout. Then $\max_x f(x) = 1$ occurs at $x = 0$.

Let $g(x) = -(1/2)x^2$. Then $\max_x g(x) = 0$ occurs at $x = 0$ and $\max_x (g(x) - f(x))$ does not exist. This prevents any version of the inequality.

What if we restrict to $f$ and $g$ where the left-hand side is defined?...

Let $g(x) = -x^4$. Then $\max_x g(x) = 0$ occurs at $x = 0$ and $\max_x (g(x) - f(x)) = -3/4$ occurs at $x = \pm 1/\sqrt{2}$. So $$ \max_x (g(x)-f(x)) \leq \max_x g(x)-\max_x f(x) $$ occurs.

Let $\displaystyle g(x) = \frac{-x^4}{8}$. Then $\max_x g(x) = 0$ occurs at $x = 0$ and $\max_x (g(x) - f(x)) = 1$ occurs at $x = \pm 1/\sqrt{2}$. So $$ \max_x (g(x)-f(x)) \geq \max_x g(x)-\max_x f(x) $$ occurs.

So no version of this inequality holds generally.