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 $argmax\Big[f+g\Big]$?


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.

  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.