Quadratic isn't that much fun!
Let
be a quadratic polynomial with real coefficients such that
for
. Find the maximum value of
.
The answer is 17.
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Let f ( x ) = A x 2 + B x + C be a quadratic polynomial for which the maximum value of the sum of the absolute value of the coefficients is achieved. Let us write f ( x ) = A ( x + B / 2 A ) 2 + C − B 2 / 4 A . If B / 2 A = − 1 / 2 , then we can enlarge ∣ A ∣ + ∣ B ∣ + ∣ C ∣ by translating the parabola so its vertex is at − 1 / 2 , and then stretch it until the y -value at the vertex is − 1 , say, and the y -value at x = 0 , 1 are + 1 . Thus we may assume that B / 2 A = − 1 / 2 , so we have f ( x ) = A x 2 − A x + C . Now, we want f ( 1 / 2 ) = − 1 and f ( 0 ) = f ( 1 ) = 1 . This implies that C = 1 , and A = 8 . Thus an acceptable choice for f is f ( x ) = 8 x 2 − 8 x + 1 .