Three Crossings Below Three

★ 8/10functions calculus algebra Polynomials Vieta's Formulas

Problem

Consider the line $l \colon kx - y - k + 1 = 0$ and

f(x) = \begin{cases} x^3 - 3x^2 + 2x + 1, & x \leqslant 2,\\ ax - 2a + 1, & x > 2. \end{cases}

The line $l$ meets the graph of $f$ at three points with $x$ -coordinates $x_1, x_2, x_3$ . If for every $0 < k < 3$ one has $x_1 + x_2 + x_3 < 3$ , find the range of possible values of the real number $a$ .

Answer

Solution

Difficulty	8/10
Topics	functions, calculus, algebra, Polynomials, Vieta's Formulas

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