9th Swedish 1969 Problem 4 Define g(x) as the largest value of |y 2 - xy| for y in [0, 1]. Find the minimum value of g (for real x).
3 - √8 Solution y 2 - xy = (y - x/2) 2 - x2/4, so y 2 - xy has a minimum at y = x/2. It is decreasing for y  x/2. Thus the largest value of y 2 - xy must be at y = 0, x/2 or 1. The values there are 0, x 2 /4, |1-x|. So for x outside the interval (0,1), g(x) ≥ 1/4. For x ∈ (0,1) we have g(x) = max(x 2 /4, 1-x). The quadratic x 2 + 4x - 4 = 0 has roots -2 ± √8, which are approx and -4.83 and 0.83. Hence g(x) = 1-x for x ≤ √8 - 2, and x 2 /4 for x ≥ √8 - 2, and the minimum value of g(x) in the interval (0,1) is g(√8 - 2) = 3 - √8, which is approx 0.18 and < 1/4. Thus this is also the minimum value of g(x) for all x. 9th Swedish 1969 © John Scholes jscholes@kalva.demon.co.uk 13 February 2004 Last corrected/updated 13 Feb 04