Finding the Maximum and Minimum Points on Graphs: Study with Video Lessons, Practice Problems & Examples

To identify maximum and minimum points on a graph, look for critical points where the graph changes direction. A maximum occurs at the peak, where the graph rises and then falls, while a minimum is at the lowest point, where it falls and then rises. Not all graphs have both; some may only have a maximum or none at all. Recognizing these points is essential for analyzing graphs effectively, as they indicate significant changes in the behavior of the function represented.

Finding Maximum and Minimum on a Graph

Alright, so now we're going to do a quick recap on how to find the maximum point on a graph and how to find the minimum point on a graph. So here in this example, I've got this kind of upside-down U, so how do we find the maximum or minimum point? So what you'll see is that not all graphs have a maximum or minimum. It's only when they kind of turn around like this, right, where they're going up, up, up, up, up, and then they turn around and start going down. Right? So when we want to find the maximum point, it's that point where it turns around. So if you notice here, the graph seems to be rising, rising, rising, rising, rising, rising, rising, and then on this side, it's falling, falling, falling. Right? So we've got to find that point where it turns around. So notice here it's still rising a little bit. Right? It's still rising a little bit, and then here it's pretty clear it's the point where it turns around. I'm going to do it in a different color there. So right here is the point where it turns around. We're not doing any math here. I just want to be able to identify the maximum and the minimum. So right here, that is our maximum. Okay? You're going to want to be able to do this, and find maximums and minimums on a graph. So what you'll notice is this one doesn't have a minimum. It went up up up to a point and then started going down down down. There wasn't a point where it was at a max bottom or max top. You might think this is a minimum here, this is a minimum here, but usually, these graphs are going to continue. So it would continue going down and it would be, you know, there wouldn't really be a minimum. So, in this case, when we see a critical point, it's kind of where it's turning around there, not where it just stops. Right? So that will be our critical point for our maximum, that we might want to identify. Let's do the same thing with a minimum point right here. So I'm thinking you guys can guess where the minimum point is going to be, but let's go ahead and do the same kind of method here. You see that the graph is falling and falling and falling, right, and then on the other side, it starts rising again. So there had to be a point where it turned around. It was falling for a while then now it's rising. Where did it turn around? It's right here. That is our minimum point, and for now all we want to do is be able to recognize when a graph has a minimum or a maximum, and then later on, we will be able to use this information when we're analyzing graphs. Cool? Alright. Let's move on.

Here’s what students ask on this topic:

How do you find the maximum point on a graph?

To find the maximum point on a graph, look for the highest point where the graph changes direction from rising to falling. This is known as a critical point. For example, if the graph is increasing (rising) and then starts decreasing (falling), the point where this change occurs is the maximum point. Mathematically, you can find this by taking the derivative of the function and setting it to zero to find critical points, then using the second derivative test to confirm if it is a maximum.

How do you find the minimum point on a graph?

To find the minimum point on a graph, look for the lowest point where the graph changes direction from falling to rising. This is another type of critical point. For instance, if the graph is decreasing (falling) and then starts increasing (rising), the point where this change occurs is the minimum point. Mathematically, you can find this by taking the derivative of the function and setting it to zero to find critical points, then using the second derivative test to confirm if it is a minimum.

What is the difference between a maximum and a minimum point on a graph?

A maximum point on a graph is the highest point where the graph changes direction from rising to falling, indicating a peak. A minimum point is the lowest point where the graph changes direction from falling to rising, indicating a trough. Both are critical points, but a maximum represents the highest value in a local region, while a minimum represents the lowest value in a local region.

Can a graph have more than one maximum or minimum point?

Yes, a graph can have more than one maximum or minimum point. These are known as local maxima and minima. A local maximum is a point where the function value is higher than all nearby points, and a local minimum is a point where the function value is lower than all nearby points. However, there can only be one global maximum and one global minimum, which are the highest and lowest points on the entire graph, respectively.

What is a critical point on a graph?

A critical point on a graph is a point where the derivative of the function is zero or undefined. These points are significant because they can indicate where the graph changes direction, leading to potential maximum or minimum points. To determine if a critical point is a maximum or minimum, you can use the second derivative test. If the second derivative is positive at the critical point, it is a minimum; if negative, it is a maximum.