Chapter 5, Problem 8SGR

To determine

Whether the statement “The relative maximum and the relative minimum of the function are often referred to $\text{end behavior}$ .” is true or false.

Answer to Problem 8SGR

The statement “The relative maximum and the relative minimum of the function are often referred to $\text{end behavior}$ .” is false.

Explanation of Solution

Given information:

The statement “The relative maximum and the relative minimum of the function are often referred to $\text{end behaviour}$ .”

Consider the provided statement “The relative maximum and the relative minimum of the function are often referred to $\text{end behavior}$ .”

The relative maximum and the relative minimum of the function are often referred to $\text{turning points}$ .

The $\text{end behavior}$ of a polynomial function is a behavior of a graph of $f\left(x\right)$ as $x$ approaches positive or negative infinity. So the sign of the leading coefficient is sufficient to predict the end behavior of a function.

For example: Degree of a polynomial function. $x^4-4x^3+3x+25$ . Find the end behavior of the equation.

Here the solution of the equation is:

The degree of the function is even and the leading coefficient is positive. So, the end behavior is:

  $\begin{array}{l}f\left(x\right)\to +\infty ,asx\to -\infty \\ f\left(x\right)\to +\infty ,asx\to +\infty \end{array}$

Thus, the statement “The relative maximum and the relative minimum of the function are often referred to $\text{end behavior}$ .” is false.