# Second derivative test for functions of 2 variables

1. Oct 2, 2011

### funzsquare

urgent! second derivative test for functions of 2 variables

1. The problem statement, all variables and given/known data
f(x,y)=x^4 - y^2 - 2x^2 + 2y - 7

2. Relevant equations
classify points (0,1) and (-1,1) as local maximum, local minimum or inclusive

3. The attempt at a solution

f(x,0)=4x^3 - 0 - 4x + 0 - 0 = 4x^3-4x
f'(x,0)=12x^2-4
f''(x,0)=24x

f(0,y)=0 - 2y - 0 + 2 - 0 = -2y+2
f'(0,y)=-2
f''(0,y)=0

how to find the points as above? as i am stuck.

2. Oct 2, 2011

### Kreizhn

Re: urgent! second derivative test for functions of 2 variables

Why are you fixing x=0 and y=0 when you calculate the derivates? You need to use the Hessian matrix to solve your problem.

3. Oct 2, 2011

### funzsquare

Re: urgent! second derivative test for functions of 2 variables

so i use the 2x2 matrix?
do i equate the 4x^3-4x=0 & -2y+2=0?

sry as i am new just start learning.

4. Oct 2, 2011

### Kreizhn

Re: urgent! second derivative test for functions of 2 variables

No worries. Slow down though. Why are you equating anything to zero? Can you tell me what the definition of the Hessian is? If you can, calculate the components and put them into the matrix. After that, tell me how we can use the Hessian to determine whether a point is maximal, minimal, or a saddle?

5. Oct 2, 2011

### funzsquare

Re: urgent! second derivative test for functions of 2 variables

i thought of finding the stationary point.
square matrix of second-order partial derivatives.
fx=4x^3-4x
fy=-2y+2
is this correct?

6. Oct 2, 2011

### Kreizhn

Re: urgent! second derivative test for functions of 2 variables

Yep. But now you need fxx fxy and fyy.

7. Oct 2, 2011

### Kreizhn

Re: urgent! second derivative test for functions of 2 variables

You can find the stationary points if you want, but I believe they've already been given to you.

8. Oct 2, 2011

### funzsquare

Re: urgent! second derivative test for functions of 2 variables

fxx=12x^2-4
fyy=-2
fyx=0
fxy=0

correct?

Last edited: Oct 2, 2011
9. Oct 2, 2011

### Kreizhn

Re: urgent! second derivative test for functions of 2 variables

Throw a negative sign in front of fyy and that will be correct. Now, how can you determine whether or not (0,1) or (-1,1) are min/max/saddle using the Hessian?

10. Oct 2, 2011

### funzsquare

Re: urgent! second derivative test for functions of 2 variables

[12x^2-4 0]
-2 0

do i need to find the eigenvalues?

11. Oct 2, 2011

### funzsquare

Re: urgent! second derivative test for functions of 2 variables

0,1 inconclusive -1,1 maximum?

12. Oct 2, 2011

### funzsquare

Re: urgent! second derivative test for functions of 2 variables

anyone?

13. Oct 2, 2011

### funzsquare

Re: urgent! second derivative test for functions of 2 variables

up!!

14. Oct 2, 2011

### Kreizhn

Re: urgent! second derivative test for functions of 2 variables

Check your matrix again, it should be

$$\begin{pmatrix} 12x^2 & 0 \\ 0 & -2 \end{pmatrix}$$

Now you do indeed have to find the eigenvalues of this matrix for each point you want to consider. However, the eigenvalues of diagonal matrices are quite easy...

15. Oct 3, 2011

### funzsquare

Re: urgent! second derivative test for functions of 2 variables

so 0,1 maximum -1,1 inconclusive