Find maximum of a function when its integral is known

Question

A very open question: have people considered a problem where the integral of a function is known (or we know a bound on it), and the goal is to find the location of its maximum?

Does this pop up in any applications?

Answer 1

Correct me if I am wrong, but it seems you are asking "is there an application to the fundamental theorem of calculus?". You say the integral is known, so assuming sufficient regularity, you have something like

$$F(x) = \int_a^x f(y)dy, \quad a\in[-\infty,\infty]$$

and you want to maximize this, so you compute $$F'(x) = f(x)$$

So that any extrema of $F$ are simply the zeros of $f$. Or if your question is to maximize $f$ given $F$, then you look at the zeros of $$F''(x) = f'(x)$$ Let me know if I am interpreting you question correctly.

Note that one application of this is when $F$ is a cumulative distribution function and $f$ is a probability density (and $a=-\infty$). Then values in a neighborhood of the maximum of $f$ have the greatest probability of occuring.