I have $n$ variables denoted by $x_1,x_2,\ldots x_n$ all of which take values in $(0,1)$.
I am trying to minimize $f(\vec{x})=1-\prod_{i=1}^{n}(1-x_i)$ for a given $\prod_{i=1}^{n}(x_i)=k $, where $k$ is a constant.
My intuition is that the point of minimum will have $x_i=x_j$ $\forall i,j \in \{1,2,\ldots n\}$. Can someone help me verify this mathematically.