A Hidden-Zero Bound

★ 8/10functions Extrema calculus Monotonicity Estimation

Problem

Let

f(x) = \ln x - a\sqrt{x + 1} + 4.

1. When $a = \sqrt{3}$ , find the intervals of monotonicity of $f(x)$ . 2. Suppose $f(x)$ has two zeros.

Find the range of possible values of $a$ .
Prove that

f(x) < \frac{2}{\sqrt{a^2 + 1} - 1} \qquad \text{for all } x \text{ in the domain}.

Answer

Solution

Difficulty	8/10
Topics	functions, Extrema, calculus, Monotonicity, Estimation

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