Understanding the First Derivative Test

This video tutorial explains how to find relative extrema of a function using the first derivative test. It covers the conditions for relative minima and maxima, the steps to locate critical values, and how to determine the sign of the first derivative. A practical example is provided to illustrate the process, including using a graphing calculator to verify results.

What is the primary goal of the First Derivative Test?

• To find relative extrema of a function

• To calculate the integral of a function

• To determine the concavity of a function

• To find the absolute extrema of a function

When does a function have a relative minimum at a point C?

• When the function is decreasing to the left and increasing to the right of C

• When the function is increasing to the left and decreasing to the right of C

• When the function is decreasing to the left and right of C

• When the function is increasing to the left and right of C

What does it mean if the sign of the first derivative does not change?

• The function has neither a relative max nor min

• The function has a relative minimum

• The function is undefined

• The function has a relative maximum

What does a positive derivative indicate about a function on an interval?

• The function is increasing on the interval

• The function has a relative maximum on the interval

• The function is decreasing on the interval

• The function is constant on the interval

What is the first step in finding relative extrema using the first derivative?

• Graph the function

• Locate any critical values of the function

• Determine the sign of the second derivative

• Evaluate the original function at critical points

How do you determine the sign of the first derivative in the application example?

• By solving the original function

• By estimating from a graph

• By using the second derivative test

• By using a graphing calculator

What are the critical numbers found in the example function?

• X = 0 and X = 3

• X = 1 and X = -1

• X = 2 and X = -2

• X = 3 and X = -3

What is the relative maximum value found in the example?

• 3 at X = 1

• -4 at X = 2

• 4 at X = -2

• 0 at X = 0

What is the relative minimum value found in the example?

• -4 at X = 2

• 4 at X = -2

• 0 at X = 0

• 3 at X = 1

What is the original function used in the example?

• 3X squared minus 12

• X plus 2

• X cubed divided by 4 minus 3X

• X squared minus 3