12. Optimization of functions of two variables

12.1Local extrema¶

The only thing that’s left to be answered is how to use the Hessian matrix to determine the character of the stationary points. In order to do so, let

Now we distinguish between three possible cases:

• if $D_2 < 0,$ then $(x,y)$ is a saddle point

• if $D_2 < 0$ and $D_1 > 0,$ then $(x,y)$ is a local minimum

• if $D_2 < 0$ and $D_1 < 0,$ then $(x,y)$ is a local maximum

12.2Constrained optimization¶

In this section we are going to learn how to solve the following problem:

There are two ways in which we can solve the problem above:

• substitution method

• Lagrange multiplier method

~ Substitution method ~¶

Substitution method is extremely useful when the constraint is given in the form of a linear function.

~ Lagrange multiplier method ~¶

As we have seen in the previous problems, substitution method is very useful when the constraint is such that we can express one variable via the other, but we need a method for problems when that is not the case.