can a constant in a max be taken outside

Question

A really silly question, but can I do: $$\max_x \left( c \cdot f(x) \right) = c \cdot \max_x f(x)$$ It seems that way, since I'm just interested in the maximum value of $f(x)$ which is not influenced by a constant and it shouldn't matter if I multiply later.

Answer 1

You can do this if and only if $c \geq 0$ (assuming that the max exists and is finite, otherwise you need $c>0$). If $c<0$ then $\max \{ cf \}=c \min \{ f \}$.