Question

find the critical point(s) of f(x, y) = x^2 + y^2 + 2xy and classify them as minimum, maximum or saddle points

Answer 1

i'll get a start on it:

Fx = 2x + 2y
Fy = 2x + 2y

so Fx = 0 when y = -x so critical points lie on this line

Answer 2

anyone able to help? i get stuck on the Hessian

Answer 3

To find the critical points, find the points at which \(f_x=0 \text{ and } f_y=0\). In our case, we have \(f_x=2x+2y\text{ and } f_y=2y+2x\). Therefore, the critical points are all the points on the line \(y=-x\).

Answer 4

yep, i got that far. i can't figure out if it's a maximum or minimum

Answer 5

It can't be maximum; f(x,y) can go to infinity.

Answer 6

yes, but to find out whether it's a maximum or minimum, you find the hessian matrix

Answer 7

This test will fail here because \(f_{xx}f{yy}-[f{xy}]^2=0\). You can see it that it's minimum by writing \(f(x,y)=(x+y)^2\). That shows that f is at least \(0\), and that happens at all points where \(y=-x\).

Answer 8

see that*

Answer 9

yes, i can see it's a minimum, but how do i prove it?

Answer 10

shouldn't the test be succeeding every time?

Answer 11

No, the test is inconclusive when \(f_{xx}f_{yy}-f^2_{xy}=0\).

Answer 12

so what do you have to do if it's inconclusive?

Answer 13

I'm not sure if there's another test you can use here.

Answer 14

ok, what about f(x,y) = e^(1+x^2-y^2)?

answer says there's a saddle point at (0,0)

Answer 15

Fx = 2x*e^(1+x^2 - y^2)
Fy = -2y*e^(1+x^2 - y^2)

so Fx = 0 when x = 0 and Fy = 0 when y = 0

so critical point at (0,0)

Answer 16

Fxy = -4xy*e^(1+x^2 - y^2) so Fxy(0,0) = 0
Fxx = 4x^2*e^(1+x^2-y^2) so Fxx(0,0) = 0
Fyx = -4xy*e^(1+x^2 - y^2) so Fyx(0,0) = 0
Fyy(0,0) = 0

so determinant is once again 0, so the test is inconclusive

Answer 17

so how the hell do we know it's a saddle point?

Answer 18

This would probably help http://mathworld.wolfram.com/ExtremumTest.html

Answer 19

that's in one variable though

Answer 20

oh sorry!

Answer 21

thanks for your help anyway :)