Solve Multivariable Extrema for f(x) = x^2 + 4xy + y^2 + 6x + 8

[thshen34]

Homework Statement

f = x^2 + 4xy + y^2 + 6x + 8
Find minimum, maximum, or saddle point

Homework Equations

A = [f_xx, f_xy; f_yx, f_yy]

The Attempt at a Solution

Found the determinant to be zero ( i got [2 ,4; 4, 2] ) so i can't use the eigenvalues to determine the extrema.

now what do i do? If I am to use lagrange multipliers, how do i do that without a constraint?

 

[haruspex]

Yes, that method breaks down here. Try a change of coordinates to break the symmetry between the second order x and y terms.

 

[Karnage1993]

How is the determinant 0?

EDIT: Sorry, I forgot I can't give out full solutions.

Make sure you are finding the determinant correctly.

 

[haruspex]

How is the determinant 0?

Good point - I failed to check that claim.