Extremum of the function

Problem

Given the function \(f: x\mapsto \frac{x}{\ln(x)}\) in the domain \(\mathbb{R}^+\backslash\{1\}\), determine position and nature of the extrema of the graph of \(f\).

Solution

We first employ Sage to obtain an overview of the given function.

The extrema of the function are found by setting the derivative equal to zero. The derivative is obtained as

\[f'(x)=\frac{\ln(x)-1}{\ln(x)^2}\]

which can easily be verified by means of Sage:

Setting the derivative equal to zero yields the only extremum at \(x_0=e\) with \(f(x_0)=e\).

In order to determine the nature of the extremum, we calculate the second derivative of the function \(f(x)\)

\[f''(x) = \frac{-\ln(x)+2}{x\ln(x)^3}\]

and evaluate it at the extremum

\[f''(x_0) = \frac{1}{\mathrm{e}} > 0\]

This result can again be verified by means of Sage:

We conclude that the extremum is a minimum in agreement with the graph of the function obtained above.