Chapter 12.3, Problem 70E

To determine

Verify that the tangent line is horizontal.

Answer to Problem 70E

Tangents drawn on the function $f(x)=\frac{\ln x}{x}$ are horizontal.

Explanation of Solution

Given information:

Use the function and its derivative to determine any points on the graph of $f$ at which the tangent line is horizontal. Use a graphing utility to verify the results.

Y1y3 $f(x)=\frac{\ln x}{x}$ ,

  $f\text{'}(x)=\frac{1-\ln x}{x^2}$

Calculation:

Consider the function, $f(x)=\frac{\ln x}{x}$

The slope of the function or the derivative is $f\text{'}(x)=\frac{1-\ln x}{x^2}$

The slope of the tangent drawn on the function $f(x)=\frac{\ln x}{x}$ should be horizontal.

So, the slope will be $0$

  $\begin{array}{l}\frac{1-\ln x}{x^2}=0\\ \ln x=1\\ x=e\end{array}$

Now, $x=e$ should satisfy the function.

When, $x=e$

  $f(e)=\frac{\ln e}{e}$

  $=e^{-1}$

So, the points at which the function $f(x)=\frac{\ln x}{x}$ and tangents drawn it is horiontal are $\boxed{\left(e,e^{-1}\right)}$ .

Now to check the above point by calculator, proceed as follows.

Press button $"=Y"$ and then enter the expression in the window as given below.

  

Now, press the $"WINDOW"$ button to set the range.

  

Now, press the $"GRAPH"$ button to plot the graph as for different point .

  

Hence, it is verified that the tangents drawn on the function $f(x)=\frac{\ln x}{x}$ are horizontal.