Symmetry and a Derivative Sum

★ 7/10functions calculus Monotonicity algebra Functional Equations Symmetry

Problem

A function $f(x)$ and its derivative $f'(x)$ are defined on $\mathbb{R}$ , with $f(x)$ increasing. Write $g(x) = f'(x)$ . Suppose

f(3x - 2) + f(4 - 3x) = f(3), \qquad g(x) + g(4 - x) = 4.

Determine, with proof, which of the following are true:

1. $f(-1) = 0$ ; 2. $f(f(1)) > f(0)$ ; 3. $g(f(-1)) < g(f(1))$ ; 4. $\displaystyle\sum_{k=1}^{2023} g(k) = 4046$ .

Answer

Solution

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

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