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Singular Point

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The function

$\displaystyle f(x,y) = y $

has a global minimum at $ (0,0)$ on the curve, determined by

$\displaystyle g(x,y) = y^3 - x^2 = 0 \,. $

\includegraphics[width=0.4\linewidth]{bsp_singulaer2}

The Lagrange condition $ (f_x,f_y)+\lambda(g_x,g_y)=0$ is not satisfied:

$\displaystyle (0,1) + \lambda ( -2 x_*, 3y_*^2) \neq (0,0) \,. $

The reason is that the rank of the Jacobi matrix $ g^\prime(x_*,y_*) = (0,0)$ is not maximal; the necessary hypothesis for the characteriziation is violated. The local behavior of $ g$ is determined by higher order terms. At such singular points, the Lagrange condition cannot be applied.
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  automatisch erstellt am 26.  1. 2017