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Math Help - Partial differentiation problem

  1. #1
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    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.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Partial differentiation problem

    Quote Originally Posted by EconMath View Post
    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).
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