Let the maximum value of ( -1x )2 + ( -1x )2 for x [- 32, 1 2 ] be mn 2, where (m, n) = 1. Then m + n is equal to .
65
Let t = -1x; then -1x = 2 - t
For x [- 32, 1 2 ]: t [- 3, 4 ]
E = t2 + ( 2 - t )2 = 2t2 - t + 24
= 2 (t - 4 )2 + 28
E decreases on [- 3, 4 ], so it is maximum at t = - 3:
E = 29 + ( 5 6 )2 = 4 2 + 25 236 = 2936 2
m = 29, n = 36
m + n = 65
Answer: 65
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