Dynamic Profit Maximization

A clinic is evaluating the optimal capital investment to maximize profit over two periods. The number of patients treated is determined by the function p func, based on the invested capital (K). Currently, the price per treatment session is price 1 and the initial capital is K_1 = capital 1. The management is considering increasing their capital for the next period. The price per treatment session next period will be price 2. The cost associated with increasing the capital is given by cost func, and future revenues are discounted by a factor of discount. To maximize profit, how much should the clinic increase its capital? The production function gives the quantity of output, which sells at the stated price in each period. Answer with the increase in capital (K_2 minus K_1), not the new level of capital. Tomorrow's revenue is multiplied by the discount factor discount.