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Chapter 4: Problem 2

Explain how to apply the First Derivative Test.

Short Answer

Expert verified
Question: Explain the main steps in applying the First Derivative Test to find the local maxima and minima of a function. Answer: To apply the First Derivative Test, follow these steps: 1. Find the first derivative of the function by differentiating it with respect to x. 2. Find the critical points by setting the first derivative equal to zero or identifying where it is undefined. 3. Set up a number-line and divide it into intervals based on the critical points. 4. Analyze the sign of the first derivative within each interval by choosing test points. 5. Determine the local maxima and minima using the First Derivative Test rules: a local maximum occurs when the sign changes from positive to negative, a local minimum occurs when the sign changes from negative to positive, and no local extrema occur when there's no sign change.

Step by step solution

01

Find the first derivative of the function

To apply the First Derivative Test, we first need to find the first derivative of the given function, which we'll denote by f'(x). You can find this by differentiating the function with respect to the variable x, using known differentiation rules and techniques.
02

Find the critical points of the function

Once we have the first derivative, we need to find the critical points of the function. These are the points where the first derivative is equal to zero or is undefined. To do this, you can set the first derivative equal to zero and solve for x, or find the points where the derivative is undefined due to, for example, division by zero.
03

Set up a number-line and test intervals

After finding the critical points of the function, set up a number-line and divide it into intervals based on the critical points. Make sure to include the critical points themselves when dividing the number line.
04

Analyze the sign of the first derivative in each interval

In this step, you will test the intervals you created in Step 3 by finding the sign of the first derivative within each of these intervals. To do this, choose a test point within each interval, and then substitute this point into the first derivative. If f'(x) is positive in the interval, it means the function is increasing in that interval, and if f'(x) is negative, it means the function is decreasing in that interval.
05

Determine local maxima and minima using the First Derivative Test

Lastly, analyze the results obtained in the previous steps to determine the local maxima and minima of the function. According to the First Derivative Test: - If the sign of f'(x) changes from positive to negative at a critical point (x=c), then the function has a local maximum at x=c. - If the sign of f'(x) changes from negative to positive at a critical point (x=c), then the function has a local minimum at x=c. - If there's no sign change in f'(x) at a critical point (x=c), then the function doesn't have a local maximum or minimum at x=c. By applying these rules to the intervals and critical points found earlier, you can determine the local maxima and minima of the function.

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