Let $f$ be a continuous nonincreasing function on $[0,1]$ with $f(1)=0$ and $\int_0^1 f(x)dx=1$. Does there exist a constant $k$ for which we can always draw a rectangle with area at least $k$, with sides parallel to the axes, in the area bounded by the two axes and the curve $f$?
If we choose the bottom-left corner to be at the origin and the bottom-right corner at $x$, the height of the rectangle is $f(x)$. So we want to maximize $xf(x)$, which amounts to choosing $x=-f(x)/f'(x)$. But it is hard to lower-bound this quantity for an arbitrary $f$.