A Midpoint Slope Test

★ 7/10functions calculus Monotonicity Logarithms inequality

Problem

Let $f(x) = \ln x - \dfrac{a}{2}x^2 + x + 2$ with $a \in \mathbb{R}$ .

1. Discuss the monotonicity of $f(x)$ . 2. Let $g(x) = f(x) + bx - 2$ with $a \neq 0$ , and suppose $g(x)$ has two zeros $x_1 \neq x_2$ . Prove that

g'\!\left(\frac{x_1 + x_2}{2}\right) < 0.

Answer

Solution

Difficulty	7/10
Topics	functions, calculus, Monotonicity, Logarithms, inequality

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