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GRE Math: Study Guide & Test Prep27 chapters | 182 lessons | 16 flashcard sets

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Lesson Transcript

Instructor:
*Laura Pennington*

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

Max and min problems show up in our daily lives extremely often. In this lesson, we will look at how to use derivatives to find maxima and minima of functions, and in the process solve problems involving maxima and minima.

Let's imagine you own a company, and your company's profit can be modeled by the function *P*(*x*) = -10*x*2 + 1760*x* - 50000, where *P*(*x*) is your company's profit, and *x* is the number of products sold. Of course, you want your profit to be as large as possible, so you want to know how many products you should sell to make this happen. In other words, you want to know how many products you need to sell in order to maximize your profit. To find that maximum profit and solve problems similar to this one, we need to be familiar with maximum and minimum points of a function.

A **maximum point of a function** is the highest point on the graph of a function, or the point that takes on the largest *y*-value. The **minimum point of a function** is the lowest point on the graph of a function, or the point that takes on the smallest *y*-value. Now take a look at the graph below.

As you can see from the position of the lines on the graph, we have a maximum point and a minimum point, but also points referred to as ''global'' and ''local.'' The maximum or minimum point in a given interval of *x*-values is called a **local maximum** or **local minimum**, respectively. The maximum or minimum point of the whole function is called the **global maximum** or **global minimum**, respectively.

Looking back at our profit example, since we want to maximize profit, we want to find the global maximum of the function. This point is shown in the image of the graph of the function *P*(*x*) = -10*x* 2 + 1760*x*-50000 that's below.

Let's take a look at our earlier image of local and global maxima and minima.

What do you notice about where the function has a maximum or minimum? Do you see how the graph flattens out at these points? If not, you should see it now that it's been pointed out, and that's great since this is the key to finding the maxima and minima of a function using derivatives.

The **derivative** of a function tells us the slope of the function at any given point. Since the graph flattens out at the maximum and minimum points, let's think about what the slope of the function would be at these points. Ah-ha! The slope of the function would be 0, since the slope of a flat line is 0. This tells us that the derivative of the function would be zero at any maximum or minimum point. This information gives us the following steps to finding maxima and minima using derivatives.

- Find the derivative of the function.
- Set the derivative equal to 0 and solve for
*x*. This gives you the*x*-values of the maximum and minimum points. - Plug those
*x*-values back into the function to find the corresponding*y*-values. This will give you your maximum and minimum points of the function.

Let's take our profit example through these steps to solve the problem. First thing we want to do is find the derivative of *P*(*x*) = -10*x*2 + 1760*x* - 50000, which is *P* ' (*x*) = -20*x* + 1760. Next, we set this equal to zero and solve for *x*.

-20x + 1760 = 0 |
Subtract 1760 from both sides |

-20x = -1760 |
Divide both sides by -20 |

x = 88 |
This is the x-value of your maximum point |

We now know that we need to sell 88 products to maximize your profit. The last step is to plug *x* = 88 into the function to find the corresponding *y*-value, or in this instance, the maximum profit.

As you can see, when *x* = 88, *P*(88) = 27440, the maximum point of the function *P*(*x*) = -10*x*2 + 1760*x* - 50000. Therefore, we know that we need to sell 88 products to get a maximum profit of $27,440.

We know now how to find maxima and minima, but how do we know whether the points we've found are maxima or minima? In our profit example, it was implied in the problem, but that won't always be the case. Thus, we have what's called the **second derivative test** to identify maxima and minima. The second derivative test is illustrated as follows:

For example, suppose we didn't know whether we were looking for a maximum or minimum of our function *P*(*x*) = -10*x*2 + 1760*x*-50000. We know that the point (88, 27440) is a maximum or minimum, but we don't know which. To figure this out, we find the second derivative of the function *P* and evaluate it at *x* = 88. We found the first derivative to be *P* ' (*x*) = -20*x* + 1760, so the second derivative is *P* '' (*x*) = -20, and *P* ' ' (88) = -20. We see the second derivative evaluated at *x* = 88 is less than zero, so by the second derivative test, we know that (88, 27440) is a maximum point of *P*.

Let's take a few moments to review what we've learned about solving min-max problems. Specifically, we learned that a **maximum point** of a function is the highest point on the graph of a function, or the point that takes on the largest *y*-value. The **minimum point** of a function is the lowest point on the graph of a function, or the point that takes on the smallest *y*-value. In addition, we learned that the maximum or minimum point in a given interval of *x*-values is called a **local maximum or local minimum**, respectively, and that the maximum or minimum point of the whole function is called the **global maximum or global minimum**, respectively.

We also learned that we can use **derivatives**, which tell us the slope of the function at any given point, to find maximum and minimum points of a function. First, we find the points that are maxima and minima using the following steps.

- Find the derivative of the function.
- Set the derivative equal to 0 and solve for
*x*. - Plug the value you found for
*x*into the function to find the corresponding*y*value. This is your maximum or minimum point.

To determine if the point is a maximum or minimum, we use the **second derivative test**. To do this, we find the second derivative of the function, and then plug in the *x*-value of the points we found to be maxima or minima. Then we use the following rules to determine if the point is a maximum or minimum.

Obviously, this process is extremely useful, since max and min problems come up so often in daily life.

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GRE Math: Study Guide & Test Prep27 chapters | 182 lessons | 16 flashcard sets

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