Cosines, Three Ways

★ 8/10functions trigonometry Polynomials Trigonometric Identities inequality Casework

Problem

Let $a, b, c$ be real numbers and

f(x) = a\cos x + b\cos 2x + c\cos 3x.

1. When $b = 1$ and $c = 0$ , find the minimum value of $f(x)$ . 2. If $f(x) \geqslant -1$ for all $x$ , find the maximum value of $a + b + c$ and all triples $(a, b, c)$ attaining it.

Answer

Solution

Difficulty	8/10
Topics	functions, trigonometry, Polynomials, Trigonometric Identities, inequality, Casework

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