Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Mistake in the following theorem

  1. Apr 17, 2005 #1
    Does somebody see the mistake in the following theorem :

    Let suppose we want to find a sufficient condition to show the point [tex](x_0,y_0)[/tex] is a local minimum of f(x,y)

    let [tex] x=p(t),\quad y=q(t),\quad p(0)=x_0,\quad q(0)=y_0, \quad p''(0)=q''(0)=0, \quad (p'(0),q'(0))\neq0[/tex]

    Then : [tex] g(t)=f(p(t),q(t)) [/tex]...suppose g(0) is a local minimum :

    [tex] g'(0)=f_xp'(0)+f_yq'(0)=0 [/tex]

    This gives when taking p'(0)=0 or q'(0)=0 (but not both), that [tex] (x_0,y_0)[/tex] is a stationary point.

    [tex] g''(0)=f_{xx}p'(0)^2+2f_{xy}p'(0)q'(0)+f_{yy}q'(0)^2>0 [/tex]

    Putting p'(0)=0 gives [tex]f_{xx}>0[/tex] and :

    The latter can be seen as a polynomial in p'(0) with [tex]q'(0)\ne 0[/tex], hence which discriminant is strictly negative, implying :

    [tex] f_{xy}^2-f_{xx}f_{yy}<0 [/tex]

    at the given point, hence the point is a minimum.
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted