Using Differentiation to Find Maximum and Minimum Values

Using Differentiation to Find Maximum and Minimum Values
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  • 0:10 Extrema
  • 1:46 Finding Extrema
  • 5:05 Finding More Extrema
  • 7:47 Lesson Summary
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Lesson Transcript
Instructor: Lydia Neptune
If you are shot out of a cannon, how do you know when you've reached your maximum height? When walking through a valley, how do you know when you are at the bottom? In this lesson, use the properties of the derivative to find the maxima and minima of a function.

Extrema

At this global maximum point, the derivative will be zero
Super C Global Maximum

We know that an extrema is a maximum or minimum value on a graph. If I'm given a graph, I can point out where the extrema are. Here, I've got a global maximum value. It's larger than any other point on this entire region. Here, I've got a local maximum value. It's the largest point in that area. Here, I've got a global minimum, and I've got a local minimum. I also can't forget to include the end points on my graph. But how do I actually calculate these?

Let's consider Super C, Human Cannonball, for a second. Super C is shot up into the air, and we can graph his height as a function of time. If I want to know his maximum height, I can see it's right between when he stops going up, and he starts going down. So it's right at that point. If I put this on a graph, I find that a maximum value would be between where the derivative is positive and the derivative is negative. That's when the slope changes from going up to going down.

A local minimum is where the slope changes from going down to going up. So for a continuous function, when the derivative changes from positive to negative, the derivative is going to go through zero. At this global maximum value, the derivative will be zero at that point exactly. Similarly, here, for this local maximum value, the derivative will be zero at the very top. Super C, at the very top of his trajectory, was not going up, and he was not going down. His height as a function of time, that derivative, was zero right there.

The function and graph for the launch of Super C
Super C Launch Graph

Finding Extrema

We can use this to our advantage to find extreme values. So for some function y=f(x), the first thing we want to do is find the critical points of this function. (That is, where the derivative is equal to zero.) So we are going to find some x values where the derivative is equal to zero. The second step is that we are going to find what y is at those critical points. We are also going to find y at the end points. So we might realize that Super C reached the pinnacle of his height 1 second into his flight, but now we are going to find exactly how high that was - that's the y value in this case. The third step is that we are going to compare all of those y values that we just calculated, and we are going to see which one is the maximum value that corresponds to our global maximum. We will also see which one is our minimum value that is going to correspond to our global minimum value. All of the other critical points might be local maxima or minima, but not always.

So let's put some numbers on this. Let's look at Super C, the human cannonball. Let's say that his height as a function of time (I'm going to write y=f(x), so x here is time and y is height) - let's say that that function is -x^2 + 2x and his entire flight goes from 0 to 2. OK, so our first step in finding all of the extrema is to find the critical points, that is, where f`(x)=0. So I'm going to differentiate our f(x). The derivative of -x^2 + 2x is -2x + 2. Now I'm going to set that equal to zero, and solve for x. Well, f`(x)=0 when x=1. So I know at what point in time Super C reached the pinnacle, at x=1, but how high was he at x=1? I'm going to calculate f(x=1). I'm going to plug 1 into our original equation here. It's important that it's the original equation and not the derivative. So I have -(1)^2 + 2(1). That's -1 + 2, or just 1. His height at x=1 is 1. So this point here is the point (1, 1).

For the launch of Super C, (1,1) is the global max and (0,0) and (2,0) are global mins
Super C Extrema

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