Chapter 3.3, Problem 54E

To determine

Check whether the statement is true or false.

Answer to Problem 54E

The given statement is true.

Explanation of Solution

Given information:

The statement:

“The graph of $f\left(x\right)=\frac{1}{x}$ has no horizontal tangent.”

Formula Used:

Differential formula:

  $f\text{'}\left(x\right)=\frac{d}{dx}f\left(x\right)$

Here $f\left(x\right)$ is the function.

The given function is $f\left(x\right)=\frac{1}{x}$.

Differentiate the given function with respect to $x$.

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

It is known that for a fraction $\frac{a}{b}$ if $b=0$ the fraction $\frac{a}{b}$ become undefined.

From the above theory $x\ne 0$ . So $f\text{'}\left(x\right)$ never can be zero.

Thus, the graph of $f\left(x\right)=\frac{1}{x}$ has no horizontal tangent.

Hence, the given statement is true.