# extremum points of function of several variables

Example 1.  The function  $f(x,\,y)=x^{2}\!+\!y^{2}\!+\!1$  from $\mathbb{R}^{2}$ to $\mathbb{R}$ has a (global) minimum point  $(0,\,0)$,  where its partial derivatives  $\frac{\partial f}{\partial x}=2x$  and  $\frac{\partial f}{\partial y}=2y$  both equal to zero.

Example 2.  Also the function  $g(x,\,y)=\sqrt{x^{2}\!+\!y^{2}}$  from $\mathbb{R}^{2}$ to $\mathbb{R}$ has a (global) minimum in  $(0,\,0)$,  where neither of its partial derivatives  $\frac{\partial g}{\partial x}$  and  $\frac{\partial g}{\partial y}$  exist.

Example 3.  The function   $f(x,\,y,\,z)=x^{2}\!+\!y^{2}\!+\!z^{2}$  from $\mathbb{R}^{3}$ to $\mathbb{R}$ has an absolute minimum point  $(0,\,0,\,0)$,  since $\nabla{f}=2x\mathbf{i}\!+\!2y\mathbf{j}\!+\!2z\mathbf{k}=\mathbf{0}\,\implies% \,x=y=z=0$,  $\frac{\partial^{2}{f}}{\partial{x}^{2}}=\frac{\partial^{2}{f}}{\partial{y}^{2}% }=\frac{\partial^{2}{f}}{\partial{z}^{2}}=2>0$, and $f(0,\,0,\,0)\leq f(x,\,y,\,z)$ for all $(x,\,y,\,z)\,\in\mathbb{R}^{3}$.

Title extremum points of function of several variables ExtremumPointsOfFunctionOfSeveralVariables 2013-03-22 17:23:57 2013-03-22 17:23:57 pahio (2872) pahio (2872) 12 pahio (2872) Theorem msc 26B12 VanishingOfGradientInDomain