An Absolute-Value Crossing

★ 7/10functions Quadratic Equations Absolute Value inequality Symmetry

Problem

Let $a \in \mathbb{R}$ and

f(x) = |x^2 - 1| - x^2 + ax.

1. If $f(x)$ is even, find $a$ . 2. Suppose the graph of $f(x)$ meets the line $y = 2x$ at exactly $2$ points in the first quadrant, with $x$ -coordinates $x_1 < x_2$ . Prove that

4x_1 - 3x_2 < a - 2 < 4x_2 - 3x_1.

Answer

Solution

Difficulty	7/10
Topics	functions, Quadratic Equations, Absolute Value, inequality, Symmetry

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