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# Curvy Equation

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### Digital Equation

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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.*

Can you find a three digit number which is equal to the sum of the hundreds digit, the square of the tens digit and the cube of the units digit?

How many numbers are there less than $n$ which have no common factors with $n$?

Frosty the Snowman is melting. Can you use your knowledge of differential equations to find out how his volume changes as he shrinks?