Chapter 11.3, Problem 66E

To determine

To find: The points on the graph of $f\left(x\right)=\frac{\ln x}{x}$ at which tangent line is horizontal and verify the result with graphing utility.

Answer to Problem 66E

The points on the graph of $f\left(x\right)=\frac{\ln x}{x}$ at which tangent line is horizontal is $\left(e,e^{-1}\right)$ .

Explanation of Solution

Given information:

The function is $f\left(x\right)=\frac{\ln x}{x}$ and its derivative is $f^{\prime }\left(x\right)=\frac{1-\ln x}{x^2}$ .

Calculation:

The point at which the tangent line is horizontal, the derivative of the graph is equal to zero.

Equate the derivative to zero and simplify.

  $\begin{array}{c}{f}^{\prime }\left(x\right)=\frac{1-\ln x}{x^2}\\ \frac{1-\ln x}{x^2}=0\\ 1-\ln x=0\\ \ln x=1\end{array}$

Further simplify,

  $\begin{array}{c}\ln x=1\\ x=e^1\\ x=e\end{array}$

Substitute $e$ for $x$ in the function.

  $\begin{array}{l}f\left(x\right)=\frac{\ln x}{x}\\ f\left(e\right)=\frac{\ln e}{e}\\ f\left(e\right)=\frac{1}{e}\\ f\left(e\right)=e^{-1}\end{array}$

Therefore, the points on the graph of $f\left(x\right)=\frac{\ln x}{x}$ at which tangent line is horizontal is $\left(e,e^{-1}\right)$ .

Check the result graphically.

To graph a function $f\left(x\right)=\frac{\ln x}{x}$ and line $y=e^{-1}$ in same window, follow the steps using graphing calculator.

First press “ON” button on graphical calculator, press $\boxed{\text{Y}=}$ key and enter right hand side of the equation $y=\frac{\ln x}{x}$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{\text{LN}}\boxed{\text{X}}\boxed{)}\boxed{÷}\boxed{\text{X}}$ . Press the $\boxed{\text{ENTER}}$ key and enter right hand side of the equation $y=e^{-1}$ after the symbol ${\text{Y}}_2$ . Enter the keystrokes $\boxed{2\text{nd}}\boxed{\text{LN}}\boxed{\left(-1\right)}\boxed{1}\boxed{)}$ .

The display will show the equation,

  $\begin{array}{c}{\text{Y}}_{\text{1}}=\frac{\ln \left(\text{X}\right)}{\text{X}}\\ {\text{Y}}_2=e^\left(-1\right)\end{array}$

Now, press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph of given quadratic equation in standard window as shown in Figure (1).

  

Figure (1)