47th Putnam 1986

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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

© John Scholes
jscholes@kalva.demon.co.uk
30 Sep 1999
Last corrected/updated 25 Nov 03