Curvy Equation
This problem asks you to use your curve sketching knowledge to find all the solutions to an equation.
Problem
Sketch the graph of the function $\text{h}$, where:
$$ \text{h}(x) = \frac {\ln x} x, \quad (x>0) $$
- 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:$$n^m=m^n $$
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.
Student Solutions
Siddhant from Singapore International School in India and Wiktor and Jiali from LAE Tottenham in the UK sketched the function correctly.
Jiali found the $x$-intercept:
Siddhant wrote:
At $x=0,$ the $\dfrac{\ln{x}}{x}$ is undefined, but $0$ is not in the domain of $\text{h}(x)$
To investigate what happens as $x\rightarrow0$, Jiali substituted in some values of $x.$ Jiali also substituted in larger values of $x$ to investigate the shape of the curve:
Wiktor described what happens for large values of $x,$ using the graphs of $y=x$ and $y=\ln{x}$ to help:
Siddhant, Wiktor and Jiali all differntiated $\text{h}(x)$ using the quotient rule and Siddhant used $\text{h}'(x)$ to understand the shape of the graph. Siddhant wrote:
$$\begin{align}\text{h}'(x) &= \frac{\frac{\text{d}}{\text{d}x}\left[\ln{x}\right].x - \ln{x}.\frac{\text{d}}{\text{d}x}\left[x\right]}{x^2}\\
&=\frac{1-\ln{x}}{x^2}\end{align}$$ Stationary point:
$\dfrac{1-\ln{x}}{x^2} = 0\\ 1-\ln{x}=0\\ \ln{x}=1\\ x=e$
Therefore $\text{h}(x)$ is stationary at $x=e$
$\rightarrow$ Therefore as $x$ approaches $e,$ $\text{h}(x)$ increases
$\rightarrow$ As $x$ increases from $e,$ $\text{h}(x)$ decreases because $\text{h}'(x)$ becomes negative (and so must have a horizontal asymptote since Jiali showed it doesn't cross the $x$ axis again)
Siddhant, Wiktor and Jiali used the graph to solve the equation $n^m=m^n$. Jiali began:
Siddhant showed how the equation can be related to the graph:
$n^m=m^n$
Take $\ln$ on both sides $$\begin{align} \ln{n^m} &= \ln{m^n}\\ m\ln{n} &= n\ln{m} \\ \frac{\ln{n}}n &=\frac{\ln{m}}m\end{align}$$ We know:
$\text{h}(m) = \dfrac{\ln{m}}{m}\\ \text{h}(n) = \dfrac{\ln{n}}{n}$
Therefore, $n^m = m^n$ can be written as $\text{h}(m) = \text{h}(n)$
Using the graph, $\text{h}(x)$ is many-one after $x=1$ and is stationary at $x=e$.
Jiali and Wiktor showed this graphically. This is Wiktor's work:
Wiktor then substituted $2$ and $4$ into the equation to check that they work. Siddhant found $4$ by trial.
Well done!
Teachers' Resources
Why do this problem?
This problem uses A-level techniques to sketch an unfamiliar graph, and then uses the graph to justify that there is only one solution to a given equation. Students have to work out how the equation is linked to the graph that they have drawn and how to use the fact the we are looking for integer solutions. There is also the idea here that "proofs" do not have to be solely algebraic, graphs can be used to show how many solutions there are to a problem, or to show that there are no possible solutions.
This problem also gives students an introduction to university entrance test questions, and the sort of problem solving questions which might be asked at interviews for Mathematics (or Maths heavy) courses at university. More resources to help students prepare for a Mathematical university course can be found on the STEP Support programme.
Possible approach
Start by asking students to sketch some graphs that they already know, such as $y=\ln x$, $y=\text{e}^x$, $y=\frac 1 x$ and $y=x^3 - 3x$. Discuss the main features of these (zeros, gradients, turning points or absence there of, behaviour for large and small $x$, values for which the functions are undefined etc.). You could also ask students how they could use the gradient to show that the graph $y=\frac 1 x$ has no turning points.
Introduce the graph $y=\dfrac{\ln x} x$ and ask students what they can tell you about this graph. Collect ideas, and perhaps draw small portions of the graph showing suggested features (such as indicating a vertical asymptote as $x \to 0_+$). The behaviour as $x$ gets large and positive can be tricky to justify, but the sign of the gradient together with the sign of the function as $x \to \infty$ can be used to justify the behaviour of the function for large $x$.
Key questions
What can we consider 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?
- How can you piece these bits together to create a sketch of the graph?
Why does the question say "x>0" after the definition of the function?
How does the function $\text{h}(x)$ relate to the equation $n^m=m^n$?
Can you rearrange the equation so that $n$ is on one side an $m$ 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?
Possible support
It might be useful to review sketching graphs and using graphs to solve equations before starting this problem. For example, pupils could sketch $y=x^3 - 3x$ (including turning points!) and then use this to work out how many solutions there are to $x^3 - 3x = 5$. Pupils could also be asked to find a value of $k$ so that $x^3 - 3x = k$ has exactly two solutions. This Desmos page might be useful for demonstrating the number of solutions to $x^3-3x=k$ as k varies.
Start by sketching some simpler graphs as discussed above, and Desmos can be used to check that the main features are correct. Students may need to be reminded of the quotient rule for differentiation.
For the second part of the question, it might be helpful to ask students how they could solve $x^3 = 4$, $x^n = 5$ etc. Then you could ask them to consider $n^m = m$, after which they can try to rearrange $n^m = m^n$ to get $n$ and $m$ on separate sides. From here then is one more step to get something resembling the given function.
Alternatively, looking at the function $\text{h}(x) = \dfrac {\ln x} x$ suggests that taking logs might be useful, so students could start by considering $\ln (n^m) = \ln (m^n)$ rather than taking logs later.
Possible extension
An interesting question to ask is: Does $a=b \implies \ln (a) = \ln (b)$? Does $\ln (a) = \ln (b) \implies a=b$? What about other functions - such as $\cos (x)$?
There are several curve sketching questions on the STEP support website including Assignment 9, Assignment 13 and Assignment 22. The STEP 2 Curve Sketching module and the STEP database can be used to find more curve sketching questions.