Classifying Extrema and Saddle Points of Multivariable Functions

[Doug_West]

Homework Statement

Find and classify all relative extrema and saddle points of the function
f(x; y) = xy - x^3 - y^2.

Homework Equations

D = fxx *fyy -fxy^2

The Attempt at a Solution

I got D < 0 where D = -1 and fxx = 0, when x=0 and y=0. However I am unsure as to the conclusion I should arrive at when D < 0 but fxx = 0. I'm thinking that this is a saddle point?

Thanks for the help in advance,
Dough

 

[SammyS]

Homework Statement

Find and classify all relative extrema and saddle points of the function
f(x; y) = xy - x^3 - y^2.

Homework Equations

D = fxx *fyy -fxy^2

The Attempt at a Solution

I got D < 0 where D = -1 and fxx = 0, when x=0 and y=0. However I am unsure as to the conclusion I should arrive at when D < 0 but fxx = 0. I'm thinking that this is a saddle point?

Thanks for the help in advance,
Dough

You need to find locations where both of the 1st partial derivatives are zero.

 

[Doug_West]

yep those two pts are

x=0, y=0
and
x=1/6 y=1/12

 

[Doug_West]

evaluating pt 0,0 I get D<0 where D= -1 and fxx(0,0) = 0, so then this is a saddle point. However is it a saddle point because D<0 or because fxx = 0?

 

[hangten1]

guess ur doing mab127 too

when d < 0 its a saddle point doesn't matter what fxx is

 

[Doug_West]

haha yep :D, thanks for the help.