extremum points of function of several variables

The points where a function of two or more real variables attains its extremum values are found in the set containing the points where all first order partial derivatives vanish, the points where one or more of those derivatives does not exist, and the points where the function itself is discontinuous.

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
Canonical name	ExtremumPointsOfFunctionOfSeveralVariables
Date of creation	2013-03-22 17:23:57
Last modified on	2013-03-22 17:23:57
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	12
Author	pahio (2872)
Entry type	Theorem
Classification	msc 26B12
Related topic	VanishingOfGradientInDomain