Maximizing the Riemann sum for partitions of fixed size

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.