A Zero Beyond One

★ 6/10functions Extrema calculus Monotonicity Casework

Problem

Let $f(x) = a\ln x - x$ .

1. Discuss the monotonicity of $f(x)$ . 2. Let $0 < a < 1$ . If the function

g(x) = 4f(x) + \frac{x^2}{2a} - x

has a zero on $[1, +\infty)$ , find the range of possible values of $a$ .

Answer

Solution

Difficulty	6/10
Topics	functions, Extrema, calculus, Monotonicity, Casework

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