TITLE: argmax of sum of two functions given constraints on argmax of each of them
QUESTION [1 upvotes]: Suppose we have two single variable functions $f(x)$ and $g(x)$. Suppose further that
$$argmax\Big[f\Big]\in[a,b]\quad \&\quad argmax\Big[g\Big]\in[a,b]$$
where $a,b\in\mathbb{R}$, moreover, assume $$\max_x\Big[f(x)\Big]=\max_x\Big[g(x)\Big]$$
and also suppose that:
$$\exists\ C\ \subset[a,b]\quad s.t.\quad \forall x\in C\ ,\ \ f(x)=g(x)$$
And the above subset is the subset that contains all the points in which $f(x)=g(x)$.

can we conclude that we have the following?
$$argmax\Big[f+g\Big]\in[a,b]$$

If the above conclusion is not generally correct, what can we say about $argmax\Big[f+g\Big]$?

REPLY [2 votes]: Let $f$ be $1+\frac{\epsilon}{2}$ everywhere except $f(a) = 1+7\epsilon$ and $f(C)=1$
Let $g$ be $1-\frac{\epsilon}{2}$ everywhere except $g(b) = 1+8\epsilon$ and $g(C)=1$
Now add an extra peak somewhere outside the interval, for instance $f(c) = 1+5\epsilon$ and $g(c) = 1+6\epsilon$. $f+g$ is now maximized at $c$ which is arbitrary.
{"set_name": "stack_exchange", "score": 1, "question_id": 4125918}