In Exercises 40 through 49, assume that the required extreme value is a relative extremum.

Alternative forms of a gene are called alleles. Three alleles, designated A, B, and O, determine the four human blood types A, B, O, and AB. Suppose that p, q, and r are the proportions of A, B, and O in a particular population, so that p + q + r = 1. Then according to the Hardy-Weinberg law in genetics, the proportion of individuals in the population who carry two different alleles is given by P = 2pq + 2pr + 2rq. What is the largest value of P?