# Maximum of the sum of two functions.

I unsuccessfully looked for a similar topic to the one I am submitting to your attention. My question is the following.

Let $f:[0,\infty[^d \to \mathbb R$ and $g : [0, \infty[^d \to \mathbb{R}$ with $f,g \in \mathcal C^1$. Both $f$ and $g$ are strictly concave and admit a unique global maximum. Denote $h(x) \equiv f(x) + g(x)$. I know from other topics that it can be shown that $\max h(x) \leq \max f(x) + \max g(x)$. Can I conclude something about the existence of a maximum of $h(x)$? I know that this maximum does not need to exist in general but, given my assumptions, is there any condition I can apply to show existence? Unfortunately, I cannot use Weierstrass since my constraint set is not compact. In case, could you provide me with some reference I can look at?

Thanks to anyone who will try to address this question!

• Notice that $f(x)\to -\infty$ for $\|x\|\to \infty$ – user251257 Oct 13 '16 at 10:37
• Thank you for your comment! Would you mind to explain me how this could help please? – barmeyp86 Oct 14 '16 at 9:46
• this is also true for $h$. Consider the set $\{ x \mid h(x) \ge h(0) \}$, which is closed and bounded – user251257 Oct 14 '16 at 10:00
• Thank you again for your comment! Could you please explain me why $f(x) \to -\infty$ whenever $\lVert x \rVert \to \infty$? Any reference to this result would be much appreciated! – barmeyp86 Oct 14 '16 at 10:32
• assume $f$ is bounded below, use the fact, that the function is strictly concave and has a maximum – user251257 Oct 14 '16 at 10:46