Five Solutions, Exactly

★ 8/10functions calculus Absolute Value Monotonicity Casework

Problem

Let

f(x) = \begin{cases} \dfrac{\mathrm{e}x}{\mathrm{e}^x}, & x \le 2, \\[6pt] \dfrac{4x-8}{5x}, & x > 2. \end{cases}

The equation $f^2(x) - 3a|f(x)| + 2a^2 = 0$ has exactly $5$ real solutions. Find the range of $a$ .

Answer

Solution

Difficulty	8/10
Topics	functions, calculus, Absolute Value, Monotonicity, Casework

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