Zeros, Monotonicity, and a Bound

★ 7/10calculus Monotonicity Logarithms inequality Casework

Problem

Let $f(x) = \ln x - \mathrm{e}^{1-x}$ and $g(x) = a\left(x^2 - 1\right) - \dfrac{1}{x}$ .

1. Determine the number of zeros of $y = f(x)$ , with justification. 2. Let

h(x) = g(x) - f(x) + \frac{\mathrm{e}^x - \mathrm{e}x}{x\mathrm{e}^x}.

Discuss the monotonicity of $h(x)$ . 3. If $f(x) < g(x)$ holds for all $x \in (1, +\infty)$ , find the range of possible values of the real number $a$ .

Answer

Solution

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

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