Age 16 to 18
Challenge Level
Sketch the graph of the function $\text{h}$, where:
$$ \text{h}(x) = \frac {\ln x} x, \quad (x>0) $$
Some things you could think about when sketching a graph:
- Are there any values of $x$ for which the function is undefined?
- What happens as $x$ gets really large?
- What happens as $x$ gets close to 0?
- Can you find the gradient of the function? What does this tell you?
Hence, or otherwise, find all pairs of distinct positive integers $m$ and $n$ which satisfy the equation:$$n^m=m^n $$
"Hence" means that the previous part of the question should be useful in some way.
Does the function $\text{h}(x)=\frac {\ln x} x$ suggest anything you could do to the equation $n^m=m^n$?
Can you rearrange the equation so that $n$ is on one side and $m$ is on the other?
If you wanted to find two values such that $\text{f}(a) = \text{f} (b)$, how could you use a graph of $y=\text{f}(x)$ to help?
Does the function $\text{h}(x)=\frac {\ln x} x$ suggest anything you could do to the equation $n^m=m^n$?
Can you rearrange the equation so that $n$ is on one side and $m$ is on the other?
If you wanted to find two values such that $\text{f}(a) = \text{f} (b)$, how could you use a graph of $y=\text{f}(x)$ to help?
STEP Mathematics I, 1988, Q1. Question reproduced by kind permission of Cambridge Assessment Group Archives. The question remains Copyright University of Cambridge Local Examinations Syndicate ("UCLES"), All rights reserved.
Copyright © 1997 - 2023. University of Cambridge.
All rights reserved.
NRICH is part of the family of activities in the Millennium Mathematics Project.
NRICH is part of the family of activities in the Millennium Mathematics Project.