I have the minimization problem
minimize $\displaystyle f_0 = \sum_{i=1}^{N} \mu_i \left( \left( 2^\frac{R_i}{\mu_i} - 1 \right) \right)$
with constraint
$\displaystyle\sum_{i=1}^{N} \mu_i = 1$
and believe that the solution is
$\displaystyle \mu_i = \frac{R_i}{\sum_{j=1}^{N} R_j}$.
The first derivative is $\displaystyle f_0^\prime = \sum_{i=1}^{N} \left( 2^\frac{{R}_i}{\mu_i} - \frac{{R}_i}{\mu_i} \rm{log}(2)2^\frac{R_i}{\mu_i} - 1 \right)$
$N$ and $R_i$ are given. I can show that for $i=2$, $f_0^\prime = 0$, by inserting $\mu_2 = 1 - \mu_1$ in $f_0$, taking the derivative and then inserting the solution.
But how can I show it for any $i$?
I couldn't just insert the solution into the first derivative, so I tried to argue that if it works for $i=2$ it would also work for higher $i$. Or that the problem can always be solved for $i=2$ and then subdivided further. But I am not sure.
Any hints are appreciated.