Step by step
The question
Question: find and classify the critical points of f(x, y) = xy.
1 Find the critical points
At a top, a bottom or a saddle, the surface is flat: both partial derivatives are 0.
Term by term.
In xy, y is fixed and waits in front.
x gives 1.
So y.
So ∂f/∂x = y.
Term by term.
In xy, x is fixed and waits in front.
y gives 1.
So x.
So ∂f/∂y = x.
Both partial derivatives must be 0.
Call them ① and ②.
② gives x = 0 straight away.
① has only y in it.
So y = 0.
So there is one critical point: (0, 0).
2 The second partials
Term by term.
y has no x: it is a fixed number, so it gives 0.
Drop the zeros: f_xx = 0.
Term by term.
x has no y: it is a fixed number, so it gives 0.
Drop the zeros: f_yy = 0.
Term by term.
y gives 1.
So f_xy = 1.
The test number: D = f_xx × f_yy − (f_xy)².
3 Test each point
At (0, 0): f_xx is 0, f_yy is 0, f_xy is 1.
So D is −1.
Careful!
f_xx alone is not enough.
If D is negative, it is a saddle, whatever f_xx is.
D is negative: up one way, down the other.
A saddle point.
4 Check
Let's check.
Here is the surface near (0, 0).
It is a saddle: up one way, down the other.
Now small steps away from the point.
Near (0, 0), one step gives a bigger value, the other a smaller one.
That fits a saddle point.
Right!
Let's look back.
First: find the critical points.
Then: the second partials.
Then: test each point.
Last: check.