As I am doing again some elementary maths (for teaching), I have this following problem regarding Riemann sums.
Let's say we consider a function $\,f$ on $[0,1]$ and we only consider partitions of size $n$ ($n$ is a positive integer) of the interval $[0,1]$. What is the partition $x_0,x_1, x_2,..., x_n$ (with $x_0 = 0$ and $x_n = 1$) that maximizes the left Riemann sum of $\,f$ ? Is it unique ?
If $\,f$ is a linear function, I have proven that the equidistant partition is the only one, by showing the concavity of the function ad hoc (with definite positiveness of its Hessian).
But I have no remote idea about this question for the others functions.