Let $f(x)$ defined on $[0,a]$ with the following property.
- $f(0) = b$, $f(x)\geq 0$
- $f(x)$ is a continuous and non-increasing function
Let $g(x) = xf(x) + k(ab - \int_0^a f(x) dx)$.
Does there exist a $m$, such that $\max(g(x)) \geq mab$ for every $f$? If so, how does $m$ related to $k$? I'm assuming $m = h(k)$ for some function $h$ independent of $a$ and $b$, but I might be wrong.
Graphically this means, given a function $f(x)$
The area of $x f(x)$ and the area between $h(x)=b$ and $f(x)$ times $k$ is always larger than the area of the rectangle times $m$.
I don't know what area of mathematics deals with this kind of problem. Please retag if this is not analysis related.
$k=0$ then $m=0$ is the only thing I know. I have problem even proving the case when $k = 1$. I conjecture it's $m=3/4$ when $k=1$.