Finding the Minima, Maxima and Saddle Point(s) of Multivariable Functions

In this article, we will again look at multivariable functions. We will look at how it is possible to find either the minima or maxima of them. We will also introduce a concept, which is unique to multivariable functions, called ‘saddle points’. If you are not already comfortable with taking the derivative of multivariable functions, then I advise you to read these articles first: Derivative of Multivariable Functions and Hesse Matrices. Now, let us get started.

Finding Minima and Maxima for Ordinary Functions

Let us start by remembering how we find the minima and maxima of single-variable functions, i.e., of a given f(x). So, what does it mean to find the maxima or minima of a function? We look at the following graph:

We can see that sometimes our graph seems to reach either a maximum or minimum, i.e., that it suddenly starts to go up instead of down — or down instead of up. Before we look at how we can find them mathematically, we start by giving some definitions:

• Local Minimum: We can say that f(x) has a local minimum at x = c, when f(x) ≥ f(c) for every x in some open interval around x = c.
• Local Maximum: We can say that f(x) has a local maximum at x = c, when f(x) ≤ f(c) for every x…