jmc

Question

If 0 p 1 and 0 q 1, define F(p, q) by F(p, q) = -2pq + 3p(1-q) + 3(1-p)q - 4(1-p)(1-q). Define G(p) to be the maximum of F(p, q) over all q (in the interval 0 q 1). What is the value of p (in the interval 0 p 1) that minimizes G(p)?

Accepted Answer

Note that for a fixed value of p, F(p,q) is linear in q, which means that F(p,q) attains its maximum value either at q = 0 or q = 1. We compute that F(p,0) = 7p - 4 and F(p,1) = 3 - 5p. Hence, G(p) = (7p - 4,3 - 5p).Note that 7p - 4 = 3 - 5p when p = 712. Then G(p) = 3 - 5p for p < 712, so G(p) is decreasing on this interval. Also, G(p) = 7p - 4 for p > 712, so G(p) is increasing on this interval. Therefore, G(p) is minimized for p = 712.