Step by step
The question
Question: find and classify the critical points of f(x, y) = 4 − x² − y².
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.
4 has no x: it is a fixed number, so it gives 0.
x²: the power 2 comes down: −1 times 2 is −2.
The power goes down by 1.
So 2x.
y² has no x: it is a fixed number, so it gives 0.
Drop the zeros: ∂f/∂x = −2x.
Term by term.
4 has no y: it is a fixed number, so it gives 0.
x² has no y: it is a fixed number, so it gives 0.
y²: the power 2 comes down: −1 times 2 is −2.
The power goes down by 1.
So 2y.
Drop the zeros: ∂f/∂y = −2y.
Set ∂f/∂x = 0.
It has only x in it.
Divide by −2: x = 0.
Set ∂f/∂y = 0.
It has only y in it.
Divide by −2: y = 0.
So there is one critical point: (0, 0).
2 The second partials
Term by term.
2x gives 2.
So f_xx = −2.
Term by term.
2y gives 2.
So f_yy = −2.
Term by term.
2x has no y: it is a fixed number, so it gives 0.
Drop the zeros: f_xy = 0.
The test number: D = f_xx × f_yy − (f_xy)².
3 Test each point
At (0, 0): f_xx is −2, f_yy is −2, f_xy is 0.
So D is 4.
Careful!
f_xx alone is not enough.
If D is negative, it is a saddle, whatever f_xx is.
D is positive and f_xx is negative: a dome.
A local maximum.
There f is 4.
4 Check
Let's check.
Here is the surface near (0, 0).
It is a dome: the point is at the top.
Now small steps away from the point.
Near (0, 0), one step gives a smaller value, the other a smaller one.
That fits a local maximum.
Right!
Let's look back.
First: find the critical points.
Then: the second partials.
Then: test each point.
Last: check.