Sketch the graph of (log x) / x

by RoRi

October 31, 2015

Describe where the function

$f(x) = \frac{\log x}{x}$

is increasing, decreasing, concave, and convex. Sketch the graph of the function.

First we take the first and second derivatives,

$\begin{align*} f(x) &= \frac{\log x}{x} \\[10pt] f'(x) &= \frac{1-\log x}{x^2} \\[10pt] f''(x) &= \frac{\log (x^2) - 3}{x^3}. \end{align*}$

Then we have the following,

$\begin{align*} f'(x) &> 0 & \text{for } 0 < x < e, \\ f'(x) &< 0 & \text{for } x > e. \end{align*}$

This means that $f$ is increasing on $0 < x < e$ and decreasing for $x > e$ .

Next, we have

$\begin{align*} f''(x) &< 0 & \text{for } 0 < x < e^{\frac{3}{2}}, \\[10pt] f''(x) &> 0 & \text{for } x > e^{\frac{3}{2}}. \end{align*}$

This means that $f$ is concave on $0 < x < e^{\frac{3}{2}}$ and convex for $x > e^{\frac{3}{2}}$ .

Finally, we sketch the function,

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