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# How to Determine Maximum and Minimum Values of a Graph

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1. 0:06 Maximum and Minimum of a…
2. 2:20 Example #1
3. 2:57 Example #2
4. 3:39 Understanding Extreme Values
5. 4:50 Example #3
6. 5:42 Example #4
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### Erin Lennon

Erin has taught math and science from grade school up to the post-graduate level. She holds a Ph.D. in Chemical Engineering.

What is the highest point on a roller coaster? Most roller coasters have a lot of peaks, but only one is really the highest. In this lesson, learn the difference between the little bumps and the mother of all peaks on your favorite ride.

## Maximum and Minimum of a Roller Coaster

I really like roller coasters. I really like it, in particular, when you get right up to the top of the roller coaster just before you're going to go plummeting down to the bottom. I also like that point at the bottom where you're suddenly pulled back up, and your innards go all the way down to your feet.

Let's see if we can analyze a roller coaster. Let's take its position across the Earth as x and its height as y. So let's draw out the roller coaster height as a function of its location, x. I really like, in particular, the point at the top and the point at the bottom of this roller coaster function. So what are these points? These are what we call extrema. Extrema are extreme values, like a maximum or a minimum value. On a roller coaster, on this particular roller coaster, there's a maximum height, so it's a maximum y value, and there's a minimum height, a minimum y value. But what about these other two points, close to the beginning and close to the end? If I ignore the rest of this graph, that's a minimum value. If I ignore the rest of this graph, that's a maximum value. So what are those? Extrema can be both. Extrema can be global, or what we call absolute extrema. This is a maximum or a minimum value for the entire domain. That's the very top of the roller coaster and the very bottom of the roller coaster. But you can also have local extrema. We also call them relative extrema, and these are the maximum or minimum values in a small region. So going back to my roller coaster, I have the global maximum and the global minimum, as well as local maximum and local minimum places. You can think of these kind of like mountain ranges and valleys. The global maximum on earth is Mt. Everest. So I've got a local maximum as well. That's that little hill in the backyard where I throw all my trash.

## Example #1

Let's take a look at this example. In this example, we have, very obviously, a global minimum. It's at the very bottom of this graph. We also have two maximum values. We have this local maximum on the right-hand side and this global maximum on the left-hand side. Now keep in mind that every global maximum or minimum point is also going to be a local maximum or minimum point. The trash heap in my backyard is a local maximum, but Mt. Everest is also a local maximum. It's just local to that region.

## Example #2

Let's do an actual example, like y=sin(x) between 0 and 3pi. If I draw this out, how many maximum and minimum values do I have? I definitely have one global minimum value, down here at -1. What about a global maximum? In this case I have two global maximums. There are two points that are both equal to 1. That's okay because I can't pick one over the other. They're both the same value, so I've got two global maximum values. I also have local minimum values at the beginning and end of my range, so I can't forget those.

## Understanding Extreme Values

Let's make this a little bit more formal. A global maximum is at some x value, like x max where y, or f, at that point is greater than any other value in your region. So this is like saying the height on the top of Mt. Everest is greater than the height at any other point on Earth. The global minimum is at the x value (x minimum, let's say) where the value at that point is less than every other point on your domain. For example, the Dead Sea is lower than any other point on Earth, at least above ground. A local max is just the locally largest point, or the locally tallest point. It's where f(x) is greater than f(x) for any point around it. This is the local trash heap. The top of that trash heap is higher than any other point around that trash heap. The local minimum is the same way. It's the minimum value in some nearby area.

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