Showing that the expectation of a function of $Bin(n,p)$ is increasing and concave in $p$

Question

Consider the function \begin{align} f(p) = p\sum_{k=0}^n\binom{n}{k}p^k(1-p)^{n-k}\frac{1}{k+1}, \quad 0\leq p\leq 1. \end{align} Is there a smart way to show that $f(p)$ is increasing and concave in $p$? This can be confirmed by plotting the function.

The function can of course be written as \begin{align} f(p) = p \mathbb{E}\big[\frac{1}{X+1}\big], \end{align} where $X$ is a $Bin(n,p)$ random variable.

Answer 1

Using the answers to this question, we know that \begin{align} f(p)= \frac{1-(1-p)^{n+1}}{n+1}. \end{align} From this form, it is easy to compute $f'(p)$ and $f''(p)$, whereupon you will see that $f'(p)>0$ and $f''(p)<0$ for all $p\in (0,1)$.