A Shiftable Even Function

★ 6/10functions trigonometry Trigonometric Identities Symmetry

Problem

Let $\omega \in \mathbb{R}$ and $f(x) = (x - 6)^2 \sin(\omega x)$ . Suppose there exists a constant $a \in \mathbb{R}$ such that $f(x + a)$ is an even function. Determine which of the values

\omega = \frac{\pi}{2}, \quad \frac{\pi}{3}, \quad \frac{\pi}{4}, \quad \frac{\pi}{5}

are possible, with proof.

Answer

Solution

Difficulty	6/10
Topics	functions, trigonometry, Trigonometric Identities, Symmetry

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