The graph of Var(X) = np(1-p) versus p is a parabola over the domain . Do you know how to find the maximum turning point of a parabola ....?
Originally Posted by zorro
No
The turning point lies between the two intercepts. The intercepts are p = 0 and p = 1. So the (maximum) turning point occurs at p = 1/2. Substitute = 1/2 into Var(X) = np(1-p) to get the required result.