Critical Numbers and Extrema of a Polynomial Function

The video tutorial explains how to find and classify critical numbers of a degree five polynomial. It begins with determining the domain, followed by calculating the first derivative. The tutorial then solves for critical numbers using the quadratic formula and classifies them as relative maxima, minima, or neither. Finally, it verifies the results graphically.

What is the domain of a polynomial function?

• All real numbers

• Numbers between -1 and 1

• Only positive numbers

• Only negative numbers

What is the first step in finding critical numbers of a function?

• Calculating the integral

• Graphing the function

• Determining the domain

• Finding the second derivative

What is the greatest common factor factored out from the derivative in this problem?

• 30x

• 60x^3

• 150x^2

• Negative 30x^2

Which method is used to solve the non-factorable quadratic equation in this problem?

• Completing the square

• Synthetic division

• Graphical method

• Quadratic formula

What are the approximate critical numbers found in this problem?

• 0, -1.5, 3.5

• 1, -2.449, 4.449

• 0, -1.449, 3.449

• 1, -1.449, 2.449

How is the function behavior determined in each sub-interval?

• By finding the integral

• By using the original function

• By calculating the second derivative

• By testing the sign of the first derivative

What indicates a relative minimum in the function?

• Function has a zero derivative

• Function changes from increasing to decreasing

• Function changes from decreasing to increasing

• Function remains constant

At which x-value does the function have a relative maximum?

• x = -1.449

• x = 1

• x = 0

• x = 3.449

What is the approximate function value at the relative minimum?

• -100

• -51.665

• 1241.6

• 0

What does the graph confirm about the critical points?

• There are no critical points

• The calculated extrema are correct

• The function is always increasing

• The function is always decreasing