# Homework Help: Finding the critical points of a multivariable function

1. Nov 4, 2011

### Koi9

1. The problem statement, all variables and given/known data\
Find the critical points of the function. Then use the Second Derivative Test to classify each critical point (or state that the test fails).

f(x, y) = x3 + y4 - 21x - 18y2

3. The attempt at a solution

partial x derivative=3x^2-21=0

partial y= 4y^3-36y=0

i try to solve those for numbers, but the answer space in the question has space for 6 different critical point answers, so I am not sure where to go from here.

2. Nov 4, 2011

### SammyS

Staff Emeritus
What did you get when you solved these for numbers?

You should get two solutions for x and three solutions for y.

3. Nov 4, 2011

### Koi9

partial x = 3x^2-21=0
3x^2=21
x=sqrt(21)/3

partial y = 4y^3-36y=0
4y^3=36y
4y^2=36
= 6
y=3/2

must be missing something, because I don't really see another way to solve it.

Thanks,
Matt

Last edited: Nov 4, 2011
4. Nov 4, 2011

### Koi9

Assignment due soon, anyone help please?

5. Nov 4, 2011

### SammyS

Staff Emeritus
That should be $x=\pm\sqrt{\frac{21}{3}}=\pm\sqrt{7}\,.$

Similarly, use ± for the y solution.

Also, when you divided by y, you got rid of the y=0 solution.

6. Nov 4, 2011

### Staff: Mentor

In addition to what SammyS said, there's another error here.
Setting fy = 0 yields
4y3-36y=0
==> 4y(y2 - 9) = 0
I hope you can see that this equation has three solutions, none of which is 3/2.