BMO 1966 Problem 1 Find the greatest and least values of f(x) = (x 4 + x 2 + 5)/(x 2 + 1) 2 for real x. Solution
: 5, 0.95. f(x) = 5 - (4x 4 + 9x 2 )/(x 4 + 2x 2 + 1) <= 5, with equality iff x = 0. Put z = (1 + x 2 ), then f(x) = (z(z - 1) + 5)/z 2 . Put w = 1/z = 1/(1 + x 2 ), then f(x) = 5w 2 - w + 1 = 5(w - 1/10) 2 + 95/100 ≥ 0.95 with equality iff w = 1/10 or x = ±3. [Of course, the question could be done less elegantly but much faster using calculus, but these problems are aimed at pre-university students, who are assumed not to know calculus.] 2nd BMO 1966 © John Scholes jscholes@kalva.demon.co.uk 30 Aug 2000