Step by step
The question
Question: find and classify the critical points of f(x, y) = 2xy − x² − 2y² + 4y.
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 2xy, 2y is fixed and waits in front.
x gives 1.
So 2y.
x²: the power 2 comes down: −1 times 2 is −2.
The power goes down by 1.
So 2x.
2y² has no x: it is a fixed number, so it gives 0.
4y has no x: it is a fixed number, so it gives 0.
Drop the zeros: ∂f/∂x = 2y − 2x.
Term by term.
In 2xy, 2x is fixed and waits in front.
y gives 1.
So 2x.
x² has no y: it is a fixed number, so it gives 0.
2y²: the power 2 comes down: −2 times 2 is −4.
The power goes down by 1.
So 4y.
4y gives 4.
Drop the zeros: ∂f/∂y = 2x − 4y + 4.
Both partial derivatives must be 0.
Call them ① and ②.
Divide ① by 2: smaller numbers.
Divide ② by 2: smaller numbers.
From ②, get x alone: x = 2y − 2.
Put it into ①.
Open the bracket and add the like terms.
So y = 2.
Back into x = 2y − 2: x = 2.
So there is one critical point: (2, 2).
2 The second partials
Term by term.
2y has no x: it is a fixed number, so it gives 0.
2x gives 2.
Drop the zeros: f_xx = −2.
Term by term.
2x has no y: it is a fixed number, so it gives 0.
4y gives 4.
4 has no y: it is a fixed number, so it gives 0.
Drop the zeros: f_yy = −4.
Term by term.
2y gives 2.
2x has no y: it is a fixed number, so it gives 0.
Drop the zeros: f_xy = 2.
The test number: D = f_xx × f_yy − (f_xy)².
3 Test each point
At (2, 2): f_xx is −2, f_yy is −4, f_xy is 2.
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 (2, 2).
It is a dome: the point is at the top.
Now small steps away from the point.
Near (2, 2), 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.