Chapter 3.3, Problem 58E

To determine

To calculate: The correct value of the horizontal tangent of the function $f\left(x\right)=\left(x^2-1\right)\left(x^2+1\right)$.

Answer to Problem 58E

The correct option is (A).

Explanation of Solution

Given information:

The given function is $f\left(x\right)=\left(x^2-1\right)\left(x^2+1\right)$.

The options are (A) 0 (B) 1 (C) 2 (D) 3 (E) 4

Formula Used:

Differentiation power rule:

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

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

Calculation:

The function is $f\left(x\right)=\left(x^2-1\right)\left(x^2+1\right)$.

It is known that the algebraic formula $\left(a+b\right)\left(a-b\right)=a^2-b^2$.

Use that formula in the given function.

  $\begin{array}{c}f\left(x\right)=\left(x^2-1\right)\left(x^2+1\right)\\ =x^4-1\end{array}$

Differentiate the above function with respect to $x$ . Use the differentiation division rule.

  $\begin{array}{c}f\text{'}\left(x\right)=\frac{d}{dx}f\left(x\right)\\ =\frac{d}{dx}\left(x^4-1\right)\\ =4x^3\end{array}$

For horizontal tangent $f\text{'}\left(x\right)=0$.

Thus, $\begin{array}{c}4x^3=0\\ x=0\end{array}$

Hence, the correct option is (A).