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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you help with the solution or looking for help too?
Draft saved Draft deleted

Similar Discussions: Mistake in the following theorem
  1. Stupid Mistake? (Replies: 4)

  2. Hidden mistake (Replies: 4)

  3. Math mistakes (Replies: 4)

  4. Mistake in proof. (Replies: 5)