Entangled Functions

★ 7/10functions calculus Functional Equations Symmetry

Problem

Functions $f(x), g(x)$ on $\mathbb{R}$ have derivatives $f'(x), g'(x)$ , and

g(x) - f(3 - x) = 2, \qquad f'(x) = g'(x - 1), \qquad g(-x + 2) = -g(x + 2).

Determine, with proof, which of the following must be true:

1. $g(x + 2)$ is even; 2. $f'(x + 2)$ is odd; 3. $f(x)$ is periodic; 4. $\displaystyle\sum_{k=1}^{2024} g(k) = 0$ .

Answer

Solution

Difficulty	7/10
Topics	functions, calculus, Functional Equations, Symmetry

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