How to prove a function has no maximum I have a function:
$$p\cdot(w+6s)^2+(1-p)\cdot(w-s)^2$$
where $p\in(0,1)$, $w>0$ and $s\geq0$ is a choice variable.
I am looking for the maximum of the function with respect to $s$, but can quickly see that the maximum is undefined as the function tends towards infinity as $s$ increases.
The question is how can I proof that in a concise way?
Thanks in advance.

Suppose $s \ge 0$. You have 
$p (w+6s)^2+(1-p)\cdot(w-s)^2 \ge p (w+6s)^2 \ge p s^2$, hence $\sup_s p (w+6s)^2+(1-p)\cdot(w-s)^2 = \infty$.
