A Slope Bounded by a Root

★ 7/10Extrema calculus Monotonicity Logarithms inequality

Problem

Let $f(x) = \dfrac{1}{2}x^2 - 2x + a\ln x$ with $a > 0$ .

1. Discuss the monotonicity of $f(x)$ . 2. If $f(x)$ has two extreme points $x_1, x_2$ , prove that

\left|\frac{f(x_1) - f(x_2)}{x_1 - x_2}\right| < \sqrt{1 - a}.

Answer

Solution

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

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