Chapter 3, Problem 4RCC

a.

To determine

To define local maximum point and local minimum point of a polynomial.

Explanation of Solution

Given information:

The given statement is:

“What is meant by local maximum point or local minimum point of a polynomial”

When the y co-ordinate of a point $\left(\text{x,y}\right)$ on a graph of a function is larger than any other y co-ordinates on the graph at points close to $\left(\text{x,y}\right)$ , it is called a local maximum point.

When the y co-ordinate of a point $\left(\text{x,y}\right)$ on a graph of a function is smaller than any other y co-ordinates on the graph at points close to $\left(\text{x,y}\right)$ , it is called a local minimum point.

b.

To determine

To determine the number of local extrema a polynomial of degree n can have.

Answer to Problem 4RCC

If P is a polynomial of degree n, then the graph of P can have a maximum of $\text{n}-1$ local extrema.

Explanation of Solution

Given information:

The given statement is:

“How many local extrema can a polynomial of degree n have”

The local extrema of a polynomial can be written as:

$\text{P}\left(\text{x}\right){\text{=a}}_{\text{n}}{\text{x}}^{\text{n}}{\text{+a}}_{\text{n}-\text{1}}{\text{x}}^{\text{n}-\text{1}}\text{+}\mathrm{...}{\text{+a}}_{\text{1}}{\text{x+a}}_{\text{0}}$

If P is a polynomial of degree n, then the graph of P can have a maximum of $\text{n}-1$ local extrema.