Partial differentiation problem

Hi! I'm new here and struggling with a problem.

I have to the function f(x,y)=1/2x^{4}+3y^{2}-6xy

I'm ought to find and classify stationary points.

I'm not sure if I went about this right, but i began assuming ∂f/∂y=0 and ∂f/∂x=0. I got two equations with two unknowns:

(1) 2x^{3}-6y=0

(2) 6y-6x=0

Now I solved for x and y.

(1) y=x^{3}/3

(2) y=x

From this point I'm not sure how to proceed to find and classify stationary points. Help much appreciated.

Re: Partial differentiation problem

Quote:

Originally Posted by

**EconMath** I have to the function f(x,y)=1/2x^{4}+3y^{2}-6xy

I'm ought to find and classify stationary points.

From $\displaystyle \dfrac{x^3}{3}-x=0$ you'll get $\displaystyle x=0$ and $\displaystyle x=\sqrt{3}$ so the stationary points are $\displaystyle (0,0),(\sqrt{3},\sqrt{3})$ and $\displaystyle (-\sqrt{3},-\sqrt{3})$. For the corresponding hessian matrices you'll verify: $\displaystyle \det H(0,0)<0$ (saddle point), $\displaystyle H(\sqrt{3},\sqrt{3})=H(-\sqrt{3},-\sqrt{3})$ positive definite (in both cases local minimum).