Location of a Local Minimum, Maximum or Saddle Point

A function with one variable creates a line when plotted. A function with two variable creates a plane when plotted. Just as we determined minimum and maximum points of a line, we can find minimum, maximum and "saddle points" of a plane surface.

These can be seen below. A maximum point would be the top of a "hill" on a surface. A minimum point on a surface is the low point in a "valley" of the surface. A "saddle point" is where you have a minimum in one dimension and a maximum in the other. This creates what looks like a saddle…thus the name.

Maximum Minimum Saddle point

We can locate these points in 3 dimensions using a method similar to what we used in 2 dimensions. We take the partial derivatives of the function with respect to each of the two variables and set them equal to zero. Like this:

To find the minimum, maximum or saddle points for z, a function of x and y:

Then solve for x and y. These values (you may get more than one point) represent the minimum, maximum and/or saddle points.

Then the question is: which is it: a minimum, maximum and/or saddle point?

Here is how you find out. Calculate D.

Now, for D > 0 and < 0: It’s a maximum point.

D > 0 and > 0: It’s a minimum point.

D < 0: It’s a saddle point.

D = 0: The test fails.