# 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 $(0,\mathrm{\hspace{0.17em}0})$, 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 $(0,\mathrm{\hspace{0.17em}0})$, 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 $(0,\mathrm{\hspace{0.17em}0},\mathrm{\hspace{0.17em}0})$, since $\nabla f=2x\mathbf{i}+2y\mathbf{j}+2z\mathbf{k}=\mathrm{\U0001d7ce}\u27f9x=y=z=0$, $\frac{{\partial}^{2}f}{\partial {x}^{2}}=\frac{{\partial}^{2}f}{\partial {y}^{2}}=\frac{{\partial}^{2}f}{\partial {z}^{2}}=2>0$, and $f(0,\mathrm{\hspace{0.17em}0},\mathrm{\hspace{0.17em}0})\le f(x,y,z)$ 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 |