Two Zeros and a Power Bound

★ 9/10functions calculus Substitution Monotonicity Logarithms

Problem

Let $a$ be a constant and

f(x) = x\left(e^{ax} + a\right) + \ln x - 1.

1. Discuss the monotonicity of $f(x)$ . 2. Suppose $f(x)$ has two zeros $x_1 < x_2$ .

Find the range of possible values of $a$ .
If $x_1 \cdot x_2^m > e^{m+1}$ always holds, find the range of possible values of $m$ .

Answer

Solution

Difficulty	9/10
Topics	functions, calculus, Substitution, Monotonicity, Logarithms

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