Curvy Equation

Age 16 to 18
Challenge Level


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:
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!