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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
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