# Thread: Determining types of critical points of two variable functions

1. ## Determining types of critical points of two variable functions

How can you determine the type of critical point

f(x,y)=y*x^2+x*y^2

Has at (0,0)?

2. There is a version of the "second derivative test" that works for functions of two variables. The analog of df/dx for functions of two variables is the matrix
$\displaystyle \begin{bmatrix}\frac{\partial^2 f}{\partial x^2} & \frac{\partial^2 f}{\partial x\partial y} \\ \frac{\partial^2 f}{\partial x\partial y} & \frac{\partial^2 f}{\partial x^2}\end{bmatrix}$

If the determinant of that matrix (at the given (x,y)) is positive then either:
1) if $\displaystyle \frac{\partial^2 f}{\partial x^2}> 1$ the point gives a minimum.
2) if $\displaystyle \frac{\partial^2 f}{\partial x^2}< 1$ the point gives a maximum.

If the determinant is negative, then the point gives a saddle point.

If the determinant is 0, the "second derivative test" won't give you an answer and, having written all of that, I now notice that is the case!

So, instead, look what happens on lines through (0, 0). On the line x= 0, f(0, y)= 0 for all y and similarly, on y= 0, f(x, 0)= 0. But on the line y= x, $\displaystyle f(x,x)= 2x^3$ the graph goes up for x positive and down for x negative. That gives you the answer.

3. Thank you very much! I was wondering if that was the case. If for instance, for a different function, I took the lines you suggested, but they all gave functions in the form ax^4 would that make it a minimum, or be inconclusive?

4. When trying different lines, you are looking for "counter-examples" and counter examples can only prove something is NOT true.

Your original example gave lines that went both up and down- the "down" part was a counter example to "minimum" and the "up" part was a counter example to "maximum"- and that left only "saddle point".

If you try different lines and find that they give increasing values, that is a counter example to "maximum" but it does not follow that it must be a minimum- there might be other lines that you did not try on which the function decreases- it could still be either a maximum or a minimum. But situations in which the "second derivative test" does not apply (the determinant is 0) are rare. What do you get when you apply that test here?

5. Thanks very much!