# What are the local extrema an saddle points of #f(x,y) = x^2 + xy + y^2 + 3x -3y + 4#?

##### 2 Answers

#### Answer:

Please see the explanation below

#### Explanation:

The function is

The partial derivatives are

Let

Then,

The Hessian matrix is

The determinant is

Therefore,

There are no saddle points.

#### Answer:

Local minimum:

#### Explanation:

The group of points that include both extrema and saddle points are found when both

Assuming

So we have two simultaneous equations, which happily happen to be linear:

From the first:

Substitute into the second:

Substitute back into the first:

So there is one point where the first derivatives uniformly become zero, either an extremum or a saddle, at

To deduce which, we must compute the matrix of second derivatives, the Hessian matrix (https://en.wikipedia.org/wiki/Hessian_matrix):

Thus

All second order derivatives are uniformly constant whatever the values of

NB The order of differentiation does not matter for functions with continuous second derivatives (Clairault's Theorem, application here: https://en.wikipedia.org/wiki/Symmetry_of_second_derivatives), and so we expect that

In this two-variable case, we can deduce the type of point from the determinant of the Hessian,

A form of the test to administer is given here:

https://en.wikipedia.org/wiki/Second_partial_derivative_test#The_test

We see that the determinant is

As a sanity check for a one-dimensional function question, I usually post the graph of it, but Socratic does not have a surface or contour plotting facility suitable for two-dimensional functions, so far as I can see. So I will overplot the two functions

As

graph{(x-(y^2-6y+4))(y-(x^2+6x+4))=0 [-10, 5, -6, 7]}