A Local Minimum at One

★ 6/10functions Extrema calculus Monotonicity Casework

Problem

Let $f(x) = \dfrac{1}{2}x^2 - (a+1)x + a\ln x$ with $a \in \mathbb{R}$ .

1. If $x = 1$ is a local minimum point of $f(x)$ , find the range of possible values of $a$ . 2. Suppose $f(x)$ is increasing on its whole domain. If for all $x_1, x_2 \in [1, 4]$ with $x_1 > x_2$ the inequality

\frac{f(x_1) - f(x_2)}{x_1^2 - x_2^2} > m

holds, find the range of possible values of $m$ .

Answer

Solution

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

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