Question

Maximize

with respect to p. What value of p do you get?

Answer 1

$Let~f(p)=\log\left(\binom{5}{3}p^3(1-p)^2 \right )=\log\binom{5}{3}&plus;3\log p&plus;2\log(1-p)\\ \frac{\mathrm{d} f(p)}{\mathrm{d} p}=\frac{3}{p}-\frac{2}{1-p}=0\Rightarrow 3-5p=0~or,~p=\frac{3}{5}\\ \frac{\mathrm{d}^2 f(p)}{\mathrm{d} p^2}=-\frac{3}{p^2}-\frac{2}{(1-p)^2}<0~for~p=\frac{3}{5}.\\ Hence~at~p=\frac{3}{5},~f(p)~is~maximum.$