# Partial differentiation problem

• Oct 28th 2012, 05:38 AM
EconMath
Partial differentiation problem
Hi! I'm new here and struggling with a problem.

I have to the function f(x,y)=1/2x4+3y2-6xy

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) 2x3-6y=0
(2) 6y-6x=0

Now I solved for x and y.

(1) y=x3/3

(2) y=x

From this point I'm not sure how to proceed to find and classify stationary points. Help much appreciated.
• Oct 28th 2012, 09:33 AM
FernandoRevilla
Re: Partial differentiation problem
Quote:

Originally Posted by EconMath
I have to the function f(x,y)=1/2x4+3y2-6xy
I'm ought to find and classify stationary points.

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