Given $r, \rho, \mu, T$, all positive, it is desired to choose $u(t)>0$ to maximize

$$\mu x(T)+\int_{0}^{T} e^{-\rho t} \log u(t) d t$$

subject to $\dot{x}(t)=r x(t)-u(t), x(0)=10$.

Explain what Pontryagin's maximum principle guarantees about a solution to this problem.

Show that no matter whether $x(T)$ is constrained or unconstrained there is a constant $\alpha$ such that the optimal control is of the form $u(t)=\alpha e^{-(\rho-r) t}$. Find an expression for $\alpha$ under the constraint $x(T)=5$.

Show that if $x(T)$ is unconstrained then $\alpha=(1 / \mu) e^{-r T}$.