How to Use the Second Derivative Test

Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

In this lesson, we will learn what the second derivative test is and how to use it to analyze functions. We will examine the steps involved in using this test and then use it in a real world application.

Steps to Solve

The second derivative test uses the first and second derivative of a function to determine relative maximums and relative minimums of a function. Let's start with a whole bunch of definitions. Relative maximum points are points that have a greater y-value than the points around it, and relative minimum points are points that have a smaller y-value than the points around it.


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A function is concave up if its slope is increasing left to right. A function is concave down if its slope is decreasing from left to right. The point at which a function changes from concave up to concave down, or vice versa, is called an inflection point.


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Phew! Okay, now we're ready to discuss the second derivative test. It all revolves around two facts.

  1. For a function, f(x), if f ' (c) = 0, then f(c) has a maximum, minimum, or inflection point at c.
  2. If f ' ' (c) > 0, then f(x) is concave up at x = c, and if f ' ' (c) < 0, then f(x) is concave down at x = c.

These two facts lead to the second derivative test, which states that for a function f(x), if f ' (c) = 0, then the following three statements are true:

  1. If f ' ' (c) > 0, then f(c) is a relative minimum.
  2. If f ' ' (c) < 0, then f(c) is a relative maximum.
  3. If f ' ' (c) = 0, then we have to determine if the point is a relative maximum, relative minimum by using the first derivative test, which states the following:
  • If f ' changes from positive to negative at c, then f has a local maximum at c.
  • If f ' changes from negative to positive at c, then f' has a local minimum at c.
  • If f ' does not change signs at c, then f doesn't have a maximum or minimum at c.

Putting all of this together, we have that to use the second derivative test to determine relative maxima and minima of a function, f(x), we use the following steps:

  1. Find f ' (x) and f ' ' (x).
  2. Set f ' (x) = 0, and solve for x.
  3. Plug your solution(s) from step 2 into f ' ' (x) and use the rules set forth in the second derivative test to determine if there is a maximum or minimum point at these values.
  4. Plug the same values back into f(x) to find the actual value of the relative maxima or minima.

Well, that's a lot of information, but at least we have an outline of how to use this second derivative test to analyze functions!

Solution

To use the second derivative test to determine relative maxima and minima of a function, we use the following steps:

  1. Find f ' (x) and f ' ' (x).
  2. Set f ' (x) = 0, and solve for x.
  3. Plug your solution(s) from step 2 into f ' ' (x) and use the rules set forth in the second derivative test to determine if there is a maximum or minimum point at these values.
  4. Plug the same values (from step 2) back into f(x) to find the actual value of the relative maxima or minima.

Application

Again, this is a lot of information! When this is the case, it's always helpful to use an application to practice what we've just learned. Let's consider a cross-country ski trail. Suppose Emily is out cross-country skiing on a challenging 4-mile mountainous trail that can be modeled with the following function:

h(x) = (1/3)x3 - (5/2)x2 + 6x

where h(x) is the height above sea level (in hundreds of feet), and x is the number of miles from the start of the trail to the end of the trail.


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