Minima and Maxima

Functions have areas that go up and down again. At the points where there is a change, there is typically a very small plateau. These plateaus can be the maxima or minima (plural for maximum or minimum), depending on where they are on the graph. Together, they make up the extrema. A graph can have local and global extrema as well. The local is the maximum or minimum for a small space on the graph and the global is the maximum or minimum for the entire graph, including parts that might not be shown because the entire function is too large to draw on a single graph.

How to Find the Maxima and Minima

The extrema are the points where the function flattens out momentarily. It’s easy to see where these are on a graph, but it is necessary to find the exact numbers by determining where the slope is zero. When the slope is zero, the line is horizontal, even if it’s just for one spot before it starts to go up or down again. To find where the slope is zero, derivatives can help.

Slope and Derivatives

The slope of a function is simply the change in y divided by the change in x. On a single point, the slope is going to be zero because the point doesn’t go in any direction. However, derivatives can be used to find out what the slop of a very small area around the point is, which can help determine the slope at the point.

Understanding the derivatives rules can make it easier to determine the slope a point is in. Once the slope is known, it’s possible to determine if the point is a maximum or minimum by looking at whether the number is getting smaller or larger. This is done by doing a second derivative, this time of the slope, where the function’s slope is zero at x. When the second derivative at x is less than 0, it’s a maximum. When it’s greater than zero, it’s a minimum.

When the Second Derivative is Zero

There are many plateaus where the slope is zero on a graph, but they are not all maximum or minimum points. It’s important to know when a point is simply just a flat point on the graph (also called a saddle point) and when it’s a maximum or minimum. To tell, it’s necessary to find the first derivative of the function to get the slope and then the derivative of the slope at x. If this is above or below zero, it denotes a maximum or minimum. In some cases, however, it will be zero. When this happens, it’s a saddle point and not one of the local extrema for that graph.

With any differentiable function, it’s possible to find the minima and maxima for local parts of the graph or for the graph globally by using two sets of derivatives. This can also help determine if the point in question is a minimum or maximum, or if it’s a saddle point for that particular graph. Practicing derivatives and learning the rules for derivatives makes it much easier for a person to learn how to use them to find the slope and to find the extrema for a graph.