We are trying to solve the following problem. Giving a discrete random variable $V$, and a transformation function $\tilde{v}_i=\frac{v_i}{v_i+c\cdot \mathbb{E}(V)}$ we get a new random variable $\tilde{V}$.

We want to find $c$ which maximizes the variance of $\tilde{V}$

i.e. find the maximum of the function $f(c)=\sum_{i=1}^np_i\cdot \left(\frac{v_i}{v_i+c\cdot \mathbb{E}(V)}\right)^2 - \left(\sum_{i=1}^np_i\cdot \frac{v_i}{v_i+c\cdot \mathbb{E}(V)}\right)^2 $

Where, $p_i$ is the probability to obtain $v_i$

A solution for n=6 will be sufficient, although a general solution is also needed.

Thanks!