Find the point P on the line AB which maximizes 1/(AP + AB) + 1/(BP + AB).
Solution
wlog AB = 1. P must lie on the segment AB, for otherwise we can reduce both AP and BP by moving P closer to the segment. So take P on the segment a distance x from A. Then we want to maximize 1/(1+x) + 1/(2-x) = 3/(2+x-x2) subject to 0 ≤ x ≤ 1. But 2+x-x2 = 9/4 - (x - 1/2)2, so we minimize the quadratic by taking x = 0 or 1. In other words we maximize the required expression by taking P = A or B.
© John Scholes
jscholes@kalva.demon.co.uk
1 Nov 2003
Last corrected/updated 1 Nov 03