Functional Maximization

Question

So how do we solve a problem like this:

Find the function $s(x)$ such that $s(x)$ maximizes $$\int_0^{s^{-1}(k)} s(x) dx $$ where $x\in[0,10]$, $s(x)\in[0,1]$, and $k\in[0,1]$ ($k$ is a constant).

Thanks a lot guys. (So sry for my previous typos)

Answer 1

It seems to me that, if $s(x)$ is strictly increasing then for $x$ between 0 and $s^{-1}(k)$, we have $0<s(x)<k$.

Therefore: $$\int_0^{s^{-1}(k)}s(x)dx < \int_0^{s^{-1}(k)}kdx = k*s^{-1}(k) <= k*10$$

let $s_n(x)$, for $n\in\{1,2,..\}$ be a sequence of functions with: $$s_n(x) = k+(x-10)/n$$

Then $\int_0^{s^{-1}(k)}s(x)dx$ goes to 10*k when n goes to infinity.

So we should say there is no such strictly increasing s(x). But we can say that sup of this integral should be 10*k.