The Least Integer $m$

★ 6/10functions Extrema calculus Estimation Logarithms Casework

Problem

Let $f(x) = a\ln x + \dfrac{a-1}{x} - x$ with $a < 2$ .

1. Find the extreme values of $f(x)$ , discussing all cases. 2. Let $m \in \mathbb{Z}$ and $a = 1$ . If the inequality

f(x) < m - (x-2)\mathrm{e}^x

holds for every $x \in (0, 1]$ , find the minimum value of $m$ .

Answer

Solution

Difficulty	6/10
Topics	functions, Extrema, calculus, Estimation, Logarithms, Casework

Whiteboard

Your sketch is saved only in this browser. To share it, export your drawing as an image (whiteboard menu → Export as → PNG), then upload that image in the comments below.

Discussion

Ask questions, share alternate solutions, and use LaTeX freely.

0 comments