jmc

Question

For a polynomial p(x), define its munificence as the maximum value of |p(x)| on the interval -1 x 1. For example, the munificence of the polynomial p(x) = -x^2 + 3x - 17 is 21, since the maximum value of |-x^2 + 3x - 17| for -1 x 1 is 21, occurring at x = -1. Find the smallest possible munificence of a monic quadratic polynomial.

Accepted Answer

Let f(x) = x^2 + bx + c, and let M be the munificence of f(x). Then |f(-1)| M, |f(0)| M and |f(1)| M. These lead to align* |1 - b + c| & M, |c| & M, |1 + b + c| & M. align*Then by Triangle Inequality, align* 4M &= |1 - b + c| + 2|c| + |1 + b + c| &= |1 - b + c| + 2|-c| + |1 + b + c| & |(1 - b + c) + 2(-c) + (1 + b + c)| &= 2. align*Hence, M 12. Consider the quadratic f(x) = x^2 - 12. Then -12 x^2 - 12 12for -1 x 1, and |f(-1)| = |f(0)| = |f(1)| = 12, so munificence of f(x) is 12. Therefore, the smallest possible munificence of a monic quadratic polynomial is 12.