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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 $f(x)=g(x)$.

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

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

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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.

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  • $\begingroup$ Thanks for your answer, your answer made me realize I forgot one more condition; I kindly invite you to look at the revised version of the question. $\endgroup$
    – Jason
    May 3 '21 at 20:37

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