Derivative Symmetries

★ 7/10functions calculus Functional Equations Symmetry

Problem

Differentiable functions $f(x), g(x)$ on $\mathbb{R}$ satisfy

f(x + 3) = g(-x) + 4, \qquad f'(x) + g'(1 + x) = 0,

and $g(2x + 1)$ is an even function. Determine, with proof, which of the following are true:

1. $g'(1) = 0$ ; 2. the graph of $f(x)$ is symmetric about $x = 2$ ; 3. the graph of $f'(x)$ is symmetric about $x = 1$ ; 4. $\displaystyle\sum_{k=1}^{2023} f'(k)g'(k) = 1$ .

Answer

Solution

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

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