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1996 spain

Source:

October 12, 2012

functionalgebrapolynomialinequalitiestriangle inequality

Problem Statement

Consider the functions

f(x) = ax^{2} + bx + c

g(x) = cx^{2} + bx + a

, where a, b, c are real numbers. Given that |f(-1)| \leq 1 , |f(0)| \leq 1 , |f(1)| \leq 1 , prove that

|f(x)| \leq \frac{5}{4}

and |g(x)| \leq 2 for -1 \leq x \leq 1 .