Step by step
The question
Question: find and classify the critical points of f(x, y) = 2x³ − 6x + y² − 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.
2x³: the power 3 comes down: 2 times 3 is 6.
The power goes down by 1.
So 6x².
6x gives 6.
y² 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 = 6x² − 6.
Term by term.
2x³ has no y: it is a fixed number, so it gives 0.
6x has no y: it is a fixed number, so it gives 0.
y²: the power 2 comes down.
The power goes down by 1.
So 2y.
4y gives 4.
Drop the zeros: ∂f/∂y = 2y − 4.
Set ∂f/∂x = 0.
It has only x in it.
Move the number and divide by 6: x² = 1.
Two numbers square to 1: x = 1 or −1.
Set ∂f/∂y = 0.
It has only y in it.
Move the number: add 4 on both sides.
Divide by 2: y = 2.
Every x goes with every y: 2 critical points.
2 The second partials
Term by term.
6x²: the power 2 comes down: 6 times 2 is 12.
The power goes down by 1.
So 12x.
6 has no x: it is a fixed number, so it gives 0.
Drop the zeros: f_xx = 12x.
Term by term.
2y gives 2.
4 has no y: it is a fixed number, so it gives 0.
Drop the zeros: f_yy = 2.
Term by term.
6x² has no y: it is a fixed number, so it gives 0.
6 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 (1, 2): f_xx is 12, f_yy is 2, f_xy is 0.
So D is 24.
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 positive: a bowl.
A local minimum.
There f is −8.
At (−1, 2): f_xx is −12, f_yy is 2, f_xy is 0.
So D is −24.
D is negative: up one way, down the other.
A saddle point.
4 Check
Let's check.
Here is the surface near (1, 2).
It is a bowl: the point is at the bottom.
Now small steps away from the point.
Near (1, 2), one step gives a bigger value, the other a bigger one.
That fits a local minimum.
Right!
Near (−1, 2), 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.