An Interval of No Zeros

Problem

Let

f(x) = \frac{a}{x} + \frac{x}{a} - \left(a - \frac{1}{a}\right)\ln x.

Prove that there is a closed interval $D$ of length greater than $1$ (the length of $[m, n]$ being $n - m$ ) such that for every $a \in D$ , the function $f(x)$ has no zeros.

Answer

Solution

Difficulty	8/10
Topics	Extrema, calculus, Monotonicity

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

No comments yet.