Maximizing $ab$ Under a Global Bound

★ 6/10functions Extrema calculus Monotonicity Logarithms inequality

Problem

Let $a, b \in \mathbb{R}$ be such that

x\mathrm{e}^x - \ln x \geqslant ax^2 + b + 1 \quad \text{for all } x > 0.

When $ab$ attains its maximum value, find $\dfrac{1}{a} + 2\ln b$ .

Answer

Solution

Difficulty	6/10
Topics	functions, Extrema, calculus, Monotonicity, Logarithms, inequality

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