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Sketch the graph of (log x) / x

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,

Rendered by QuickLaTeX.com

One comment

  1. joao hanna says:

    Hello
    I guess that there is a problem with the plotted function . As log[1]=0 , the abciss scale is weird [ out of scale].

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