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Finding Minima & Maxima: Problems & Explanation

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  • 0:01 What Are Minima & Maxima?
  • 1:41 Determining…
  • 4:18 Real-World Examples
  • 5:52 Lesson Summary
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Instructor
Jennifer Beddoe

Jennifer has an MS in Chemistry and a BS in Biological Sciences.

Expert Contributor
Kathryn Boddie

Kathryn earned her Ph.D. in Mathematics from UW-Milwaukee in 2019. She has over 10 years of teaching experience at high school and university level.

One of the most important practical uses of higher mathematics is finding minima and maxima. This lesson will describe different ways to determine the maxima and minima of a function and give some real world examples.

What Are Minima and Maxima?

When you graph a function, the minimum value of that function can be found at its lowest point on the graph. It is the vertex of a graph that opens upwards. The maximum value of a graphed function can be found at its highest point on the graph, or the vertex if the graph opens downwards.

For example, in the following image, the minimum value of the function on the left is y = -5, while the maximum value of the function on the right is y = 5.

However, you do not need to look at the graph of a function to determine whether it has a maximum or minimum value—that information can be determined just by looking at the equation of a function. Equations for quadratic functions have the general form ax^2 + bx + c. If the x^2 term is positive, the equation will have a minimum. If the x^2 term is negative, the equation will have a maximum.

Do these functions have maxima or minima?

1.) 5x^2 + 3x - 1

Because the x^2 term is positive (5x^2), this function has a minimum.

2.) 6x - 2x^2

The -2x^2 means this function has a maximum.

3.) 7-2x + x^2

The function has a minimum because of the x^2.

4.) -2x^2 + 3x + 1

This function has a maximum because the x^2 term (-2x^2) is negative.

Maxima and minima have practical applications in the fields of engineering, finance, manufacturing, and many other areas.

Determining Maximum/Minimum Values

There are three methods for determining the maximum or minimum values of a quadratic equation. You can either graph the equation, or you can one of two equation forms to find the values.

Use the Graph

You can find maxima or minima visually by graphing an equation. The y-value of the vertex of the graph will be your minimum or maximum. This is especially easy when you have a graphing calculator that can do most of the work for you.

Again, using the graph from earlier, you can see that the minimum point of the graph with a minimum is at y = -5. The graph with a maximum point has that point at y = 5.

Use the y = ax^2 + bx + c Equation Form

As was discussed earlier, if your equation is in the form y = ax^2 + bx + c, the minimum or maximum depends on the sign associated with the x^2 term. You can then find the minimum or maximum by using the equation:

min or max = c - b^2 / 4a

The first step is to determine whether your equation gives a maximum or minimum. This can be done by looking at the x^2 term. If this term is positive, the vertex point will be a minimum; if it is negative, the vertex will be a maximum. Once this is determined, you can then use the second equation to find the exact point of the minima or maxima.

Example: Does the function (2x^2 + 5x - 3) have a minimum or maximum? What is it?

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Additional Activities

Further Exploration into Maxima and Minima

Finding the maximum or minimum value can be important in many real-world situations, such as maximizing area, minimizing cost, or maximizing profits. In the following examples, students will practice finding the maximum or minimum value of various real-world situations which are modeled with a quadratic function.

Examples

1. The sides of a rectangular garden are labeled below in feet. Can the area of the garden be made into a maximum area? If so, what is the maximum area?

2. The profit from selling x items in dollars is given by the equation

Is it possible to get a maximum profit? If so, what is it?

3. The cost of producing x items in dollars is given by the equation

What is the minimum cost for the company?

Solutions

1. The area of a rectangle is given by the length multiplied by the width. So the area of the garden is given by:

We can have a maximum area because a is negative. The maximum area is given by:


2. It is possible to get a maximum profit since a is negative. The maximum profit, to the nearest cent, is given by:


3. We do have a minimum cost since a is positive. The minimum cost, to the nearest cent, is given by:

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