I consider the function $H: \mathbb{R}\times\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ defined by
$$ H(x,u,p) = p(-xu + \frac{1}{2}u^2) $$
I would like to find its infimum with respect to the second variable however there is one case that I do not understand.
First, I check if the function is convex and I see that it depends on the sign of $p$. For the case where $p\geq 0$ there is no ambiguity (either the function is convex and can be minimized, or it is constant equal to $0$).
However, for the case where $p<0$ I feel that my argument is not rigorous enough but I may be wrong.
The idea is that we have
$$ H(x,u,p) = p(-x + \frac{1}{2}u)u $$
Thus, for $u$ big enough the parenthesis is strictly positive, taking the limit as $u$ goes to $\infty$ we obtain that $H$ goes to $-\infty$ and this implies that its his infimum with respect to $u$.
I would like to know if this argument is valid for the part where $p<0$ please. Thank you a lot