Partial differentiation problem

Oct 2012
1
0
Norway
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.
 

FernandoRevilla

MHF Hall of Honor
Nov 2010
2,163
1,186
Madrid, Spain
I have to the function f(x,y)=1/2x4+3y2-6xy
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).