A Frozen Variable

★ 6/10calculus Monotonicity Estimation inequality Casework

Problem

Let $f(x) = a\ln x + (x+1)^2$ with $a \neq 0$ and $x > 0$ .

1. Find the intervals on which $f(x)$ is monotonic. 2. If

f(x) - \frac{x^2}{a} \leqslant 0

holds for every $x \in [1, +\infty)$ , find the range of possible values of $a$ .

Answer

Solution

Difficulty	6/10
Topics	calculus, Monotonicity, Estimation, inequality, Casework

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