Classifying critical points

**edeffect** · May 21st 2008, 12:57 AM

Hello

Im trying to find all the critical points of this functions.

Could somebody please give me a hand?

Thanks Heaps

**Moo** · May 21st 2008, 02:21 AM

Hello,

$f(x,y)=x^2+x^2y+5y^2+2y^3$

$\frac{\partial f}{\partial x} (x,y)=2x+2xy=2x(1+y)$

$\frac{\partial f}{\partial y} (x,y)=x^2+10y+6y^2$

You have to solve simultaneously these equations :

$\left\{ \begin{array}{ll} 0&=2x(1+y) \\ 0&=x^2+10y+6y^2 \end{array} \right.$

from the first equation, you know that either x=0, either y=-1.
Then substitute in the second equation, for each possible value.

Then, calculate the Hessian matrix, and here, I don't know how much you've learnt :/

**edeffect** · May 21st 2008, 06:36 AM

Hey

sorry to bother you, but i solved the two equations simultaneously and now im not exactly sure what i need to do? Could you show your working so i can get a clear and definite way to solve this type of problem.

Thank you

**Moo** · May 24th 2008, 01:53 AM

Sorry, I didn't see your answer...

Take a look at this link : Hessian matrix - Wikipedia, the free encyclopedia
Normally, everything is explained (Note that you have to take the values of the second derivatives at the critical points).

I don't feel like typing all the tex, but I can do it by hand if you really need it

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