Demand q for product B depends on its price p and its quality (a product attribute) x: q = (a – bp + dx). Profit is q(p – c); the variable cost increases with its quality: c = x2; and a, b and d are positive constants. You wish to choose the price and attribute value that generates maximum profit. Assume there are no constraints (i. e.: any real value of p, x, and q are possible).

a. Write this problem mathematically as a standard optimization problem.
b. Write the first order necessary optimality conditions for this problem (you do not need to solve them explicitly).
c. Is this point a local maximum? Prove it.
d. Is this point a global maximum? Prove it.
e. What happens when you use the starting point [0,20] instead of [0,0]? Why does this happen?