(a) Recall the advertising problem (Exercise 5 of Section 6.1). Considering the problem as an infinite horizon discounted reward problem with discount factor \(\alpha=.9\), write the DP equation.

(b) Find the optimal value function and the optimal action at the state where the soft drink has the maximum \(30 \%\) of the market, as a function of the problem parameters \(r\) and \(c\).

(c) Let \(r=10, c=2\), and find the optimal value function and the optimal policy for the advertising problem, without resorting to successive approximations.