Second derivative test for functions of 2 variables

[funzsquare]

urgent! second derivative test for functions of 2 variables

Homework Statement

f(x,y)=x^4 - y^2 - 2x^2 + 2y - 7

Homework Equations

classify points (0,1) and (-1,1) as local maximum, local minimum or inclusive

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.

 

[Kreizhn]

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.

 

[funzsquare]

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.

 

[Kreizhn]

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?

 

[funzsquare]

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

 

[Kreizhn]

Yep. But now you need fxx fxy and fyy.

 

[Kreizhn]

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

 

[funzsquare]

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

correct?

 

[Kreizhn]

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?

 

[funzsquare]

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

do i need to find the eigenvalues?

 

[funzsquare]

0,1 inconclusive -1,1 maximum?

 

[funzsquare]

anyone?

 

[funzsquare]

up!

 

[Kreizhn]

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...

 

[funzsquare]

so 0,1 maximum -1,1 inconclusive