How to Find and Verify Stationary Points in a Differentiation Problem

[smn]

Hello,
I'm trying to complete the question below:
Consider F(x,y)=1/3x^3+y^2+2xy+2x+2y+1
Find the 2 stationary points of F and show that one of them is a minimum of F.
I've got as far as getting:
dF/dx = 2/3x+2y+2
dF/dy = 2x+2y+2
I would like to know what i need to do next (do i have to treat the above two terms as simultaneous equations?)
Any help will be appreciated
Regards
smn

 

[TD]

Be careful, if your initial function F(x,y) is correct then you've got a wrong partial derivative with respect to x. I see a x³/3 and its derivative should give x² and I don't see that term in your dF/dx. dF/dy seems to be correct.

Once you have the right partial derivatives, let them equal zero and solve them together (system of 2 equations). This gives the stationary points.

 

[smn]

Thanks for your reply,

You're correct, it should read:

dF/dx = 2/3x^2+2y+2, which i wrote down in my notes!Typo error.

When you say, solve them together, do you mean like you do with simultaneous equations?

I tried that route but I'm getting confused with what to do with the term that's a fraction.

I get:

2+2/3x^2+2y=0-----eq.1

2+2x+2y=0----------eq.2

so i assume that the 2 term cancels,as does the 2y and then i get lost.

 

[TD]

Yes, a system of equations so simultaneous equations.
By the way, (x³/3)' = x² and not 2x²/3...

Be careful with your derivatives, else the stationary points will be wrong too of course!