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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Since the term with the x^2, or a term, is positive, you know there will be a minimum point. To find it, plug the values into the provided equation:
min = c - b^2 / 4a min = -3 - 5^2 / 4*(2) min = -3 - 25 / 8 min = -3 - (3.125) min = -6.125
Use the y = a(x - h)^2 + k Equation Form
As with the last equation, if the a term in this equation is positive, there is a minimum. If it is negative, there is a maximum. Either way, the minimum or maximum point can be found at k. No equation or calculation is necessary - the answer is the value of k.
Example: Does this function have a maximum or minimum? -3(x + 1)^2 + 12
Since the a term is negative, there will be a maximum at y = 12.
1.) The number of bacteria in a refrigerated food is given by the equation y = -0.5x^2 + 25x + 800, where y = the number of bacteria and x = time. Will this equation give a maximum or minimum number of bacteria, and what will it be?
Because the a term is negative, we know there will be a maximum for this equation. To find that maximum, we can use the equation:
max = c - b^2 / 4a max = 800 - (25^2) / (4*(-0.5)) max = 800 - 625 / -2 max = 800 + 312.5 max = 1112 bacteria
And that will be the maximum amount of bacteria present.
2.) A manufacturer of tennis balls has a daily cost of y = x + 0.02x^2 + 100. Does this function have a minimum or maximum, and what is it?
Because the a term is positive, there will be a minimum. Using the equation min = c - b^2 / 4a, we can find the minimum cost.
min = 100 - (1^2) / (4*0.02) min = 100 - 1 /( 0.08) min = 100 - 12.5 min = $87.50
Let's review. You can determine if a quadratic function has a maximum or minimum by looking at either the graph of the function or the equation. When looking at the equation, pay attention to the a term. If it is positive, the function will have a minimum. If it's negative, the function will have a maximum.
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.
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?
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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