# Thread: Critical points of functions of two variables

1. ## Critical points of functions of two variables

Show that
$
\Delta=f_{xx}f_{yy}-(f_{xy})^2
$

is zero at the origin. Then classify this critical point by visualizing the surface
$
f(x, y) = x^3 + y^3
$

I have proved the first condition, but I am not sure how to proceed with the second part to classify this critical point by visualizing the surface.

Any help will be greatly appreciated!

2. You should at least be able to visualise what the cross-sections of the surface along the coordinate axes look like. On the x-axis, for example, where y=0, f(x,y) is equal to x^3, and you surely know what the graph of that curve looks like, with a point of inflection at the origin.

3. From what I am understanding the critical point then should be a saddle point, am I right?

4. Originally Posted by hasanbalkan
From what I am understanding the critical point then should be a saddle point, am I right?
No, because there is no direction in which it is either a local maximum or a local minimum. It is neither a max nor a min nor a saddle point. It is a point of inflection. In fact, the function f(x,y) is identically zero along the line y=x. It is positive above that line and negative below it.

5. Thank you a lot for the reply. I did some research and according to the results I found this critical point is called a stationary point of inflection also known as a saddle point.

Saddle Point -- from Wolfram MathWorld

Once again, great help on this forum...