# Mistake in the following theorem

1. Apr 17, 2005

### kleinwolf

Does somebody see the mistake in the following theorem :

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

let $$x=p(t),\quad y=q(t),\quad p(0)=x_0,\quad q(0)=y_0, \quad p''(0)=q''(0)=0, \quad (p'(0),q'(0))\neq0$$

Then : $$g(t)=f(p(t),q(t))$$...suppose g(0) is a local minimum :

$$g'(0)=f_xp'(0)+f_yq'(0)=0$$

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

$$g''(0)=f_{xx}p'(0)^2+2f_{xy}p'(0)q'(0)+f_{yy}q'(0)^2>0$$

Putting p'(0)=0 gives $$f_{xx}>0$$ and :

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

$$f_{xy}^2-f_{xx}f_{yy}<0$$

at the given point, hence the point is a minimum.