Two Zeros and a Bounded Minimum

★ 9/10Extrema calculus Monotonicity AM-GM

Problem

Let $a \in (0, 1)$ , and define

f(x) = x - \frac{a}{x} - (a+1)\ln x + a - 1, \qquad g(x) = \frac{x+a}{x-1}\ln x.

1. Prove that $f(x)$ has exactly $2$ zeros. 2. For $x \in (0, 1)$ , prove that $g(x)$ attains a minimum value $h(a)$ , and that $0 < h(a) < 2$ .

Answer

Solution

Difficulty	9/10
Topics	Extrema, calculus, Monotonicity, AM-GM

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