Finding the PDF that maximizes the $\alpha$-th moment

Question

Given $\alpha$ and constant $\mu$,

$$\begin{array}{ll} \text{maximize} & \displaystyle\int_0^\infty p(x)x^\alpha \,\mathrm d x\\ \text{subject to} & \displaystyle\int_0^\infty p(x)\,\mathrm d x = 1\\ & \displaystyle\int_0^\infty p(x)x \, \mathrm d x = \mu\end{array}$$