Optimal control of an integrator

Question

Given the control problem $$ \max_{u \in [0,1]} \int_0^{10} x (t) \,{\rm d} t \quad \text{subject to} \quad \dot x = u, \quad x(0)=0, \quad x(10)=2$$

a) Find the solution $(x_∗,u_∗)$ that satisfies the Maximum Principle.

b) Motivate whether the solution found is optimal or not.

c) Compute the value $V$ of the objective function corresponding to the optimal solution.