Question

Use a sign chart for $f\text{'}\text{'}$to determine the intervals on which each function f in Exercises 41–52 is concave up or concave down, and identify the locations of any inflection points. Then verify your algebraic answers with graphs from a calculator or graphing utility.

$f\left(x\right)=\frac{x}{\ln x}$

Short answer

Inflection point is $x=e^2\cong 7.4$, concave up at$\left(-\infty ,e^2\right)$and concave down at$\left(e^2,\infty \right)$.

Step-by-step solution

Step 1. Given information.

The given function is$f\left(x\right)=\frac{x}{\ln x}.$

Ste[ 2. Second Derivative.

On differentiating,

$$\begin{gathered} f^{\text{'}}\left(x\right)=\frac{d}{dx}\frac{x}{\ln x} \\ =\frac{\ln x-1}{{\ln }^{2}x} \\ {f}^{\text{'}\text{'}}\left(x\right)=\frac{d}{dx}\frac{\ln x-1}{{\ln }^{2}x} \\ =-\frac{\ln x-2}{x{\ln }^{3}x} \end{gathered}$$

Step 3. Sign chart.

Now, $f^{\text{'}\text{'}}\left(x\right)=0whenx=e^2.$

Therefore,

the chart will be,

$$\begin{gathered} InflectionPoint:x=e^2\cong 7.4 \\ Concaveup:\left(-\infty ,e^2\right) \\ Concavedown:\left(e^2,\infty \right) \end{gathered}$$

Step 4. Verification.

The graph of the function is ,