Putnam 41/A4 solution

47th Putnam 1986

Problem A1

S is the set {x real st x⁴ - 13x² + 36 ≤ 0}. Find the maximum value of f(x) = x³ - 3x on S.

Solution

Answer: 18.

Trivial.

x⁴ - 13x² + 36 = (x² - 4)(x² - 9), so S is the union of the intervals [-3, -2] and [2, 3]. But f(x) is negative on the first interval and positive on the second, so the maximum value occurs in the second interval. But f(x) is strictly increasing on the second interval (it is the product of x and x² - 3, both strictly increasing and positive), so the maximum value occurs at the endpoint x = 3 and is 18.

47th Putnam 1986