A. For the function in Exercise 15, find the average of the...

a. For the function in Exercise 15, find the average of the critical numbers. b. For the function in Exercise 15, use a graphing calculator to find the roots of the function, and then find the average of those roots. c. Compare your answers to parts a and b. What do you notice? d. Repeat part a for the function in Exercise 17. e. Repeat part b for the function in Exercise 17. f. Compare your answers to parts d and e. What do you notice? It can be shown that the average of the roots of a polynomial (including the complex roots, if there are any) and the critical numbers of a polynomial (including complex roots of $f^{\prime}(x)=0$ if there are any ) are always equal.

a. For the function in Exercise 15, find the average of the critical numbers.
b. For the function in Exercise 15, use a graphing calculator to find the roots of the function, and then find the average of those roots.
c. Compare your answers to parts a and b. What do you notice?
d. Repeat part a for the function in Exercise 17.
e. Repeat part b for the function in Exercise 17.
f. Compare your answers to parts d and e. What do you notice? It can be shown that the average of the roots of a polynomial (including the complex roots, if there are any) and the critical numbers of a polynomial (including complex roots of $f^{\prime}(x)=0$ if there are any ) are always equal.

Key Concepts

Critical Numbers

Critical numbers of a function are the values in the domain where the derivative is zero or undefined. These points are important because they may correspond to local maxima, minima, or points of inflection, helping to analyze the function’s behavior.

Roots of a Polynomial

Roots of a polynomial are the solutions to the polynomial equation when it is set equal to zero. These values indicate where the function intersects the x-axis and provide insight into the function's overall structure and behavior.

Arithmetic Mean

The arithmetic mean is the sum of a collection of numbers divided by the number of elements in the collection. This concept is used to find a central or balanced value among a set of values such as roots or critical numbers.

Relationship Between Roots and Critical Numbers

For polynomials, there exists an intriguing connection where the arithmetic mean of all roots, including complex ones, is equal to the arithmetic mean of the critical numbers. This phenomenon is related to the symmetric properties of polynomial functions and often can be understood through the application of Vieta's formulas.

Transcript

00:01 For this problem, we are going to be referencing two exercises from earlier in this chapter.

00:07 We're going to be looking at exercise number 15 and 17.

00:11 So if you're not sure where some of these numbers come from, you might want to go back and reference those two problems.

00:18 So first we're going to start with the equation from exercise 15, which is the one that i'm marking in red.

00:24 F of x equals two thirds x cubed minus x squared, minus 24x, minus 24x, minus.

00:30 As four.

00:32 Now, we found in exercise 15 that there are two critical numbers, negative three and four.

00:40 Now, what we're supposed to start with here is to find the average of these two critical numbers.

00:45 So if i average them, i add them up and divide by two, i get an average of one half.

00:53 Now, let's take a look at where the roots of this function are.

00:57 And i'm going to use, you can a graphing calculator or a graphing application on the computer like decmos, which is what i have here.

01:04 And you can see we have a cubic function.

01:07 And if i want to know where my roots are, i have three roots.

01:14 One at negative 5 .201, one at negative .168 and one at 6 .869.

01:21 So let's take those three numbers and we're going to average those.

01:25 So again, my roots occur at the points.

01:31 Negative 5 .201.

01:34 I have negative 0 .168 plus 6 .869...

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