Three Roots and a Combination

★ 7/10functions Quadratic Equations Substitution Vieta's Formulas

Problem

Let

f(x) = \begin{cases} 2x^2 - x, & x \leqslant 0,\\ -x^2 + x, & x > 0. \end{cases}

Suppose the equation $f(x) = m$ has $3$ real solutions $x_1 < x_2 < x_3$ .

1. Find the range $D$ of possible values of $m$ . 2. As $m$ varies over $D$ , find the range of possible values of $x_1 + x_2x_3$ .

Answer

Solution

Difficulty	7/10
Topics	functions, Quadratic Equations, Substitution, Vieta's Formulas

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