You may also like

Exploring Cubic Functions

Quadratic graphs are very familiar, but what patterns can you explore with cubics?

Curve Fitter

This problem challenges you to find cubic equations which satisfy different conditions.

Curve Hunter

This problem challenges you to sketch curves with different properties.

Curvy Equation

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?
You might want to think about these in a different order, for example knowing the gradient may help you work out how the function behaves for large (or small) $x$.

 

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?

 

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.