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 $\abla{f}=2x\\mathbf{i}\\!+\\!2y\\mathbf{j}\\!+\\!2z\\mathbf{k}=\\mathbf{0}\\,\\implies%\\,x=y=z=0$ , $\\frac{\\partial^{2}{f}}{\\partial{x}^{2}}=\\frac{\\partial^{2}{f}}{\\partial{y}^{2}%}=\\frac{\\partial^{2}{f}}{\\partial{z}^{2}}=2>0$ , and $f(0,\\,0,\\,0)\\leq f(x,\\,y,\\,z)$ for all $(x,\\,y,\\,z)\\,\\in\\mathbb{R}^{3}$ .

单词	ExtremumPointsOfFunctionOfSeveralVariables
释义	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 $\abla{f}=2x\\mathbf{i}\\!+\\!2y\\mathbf{j}\\!+\\!2z\\mathbf{k}=\\mathbf{0}\\,\\implies%\\,x=y=z=0$ , $\\frac{\\partial^{2}{f}}{\\partial{x}^{2}}=\\frac{\\partial^{2}{f}}{\\partial{y}^{2}%}=\\frac{\\partial^{2}{f}}{\\partial{z}^{2}}=2>0$ , and $f(0,\\,0,\\,0)\\leq f(x,\\,y,\\,z)$ for all $(x,\\,y,\\,z)\\,\\in\\mathbb{R}^{3}$ .
随便看	table of integer factorizations for 0<n<1001 TableOfIntegerFactorizationsFor0N1001 TableOfMersennePrimes table of Mersenne primes TableOfMultiplicationUpTo12 table of multiplication up to 12 TableOfNaturalLogarithms table of natural logarithms TableOfPartialFractionExpansions table of partial fraction expansions table of polite number representations for 1<n<101 TableOfPoliteNumberRepresentationsFor1N101 TableOfPrimesBetweenTheFirst100Squares table of primes between the first 100 squares TableOfPrimesInArithmeticProgressionsPerDirichletsTheorem table of primes in arithmetic progressions per Dirichlet’s theorem TableOfPseudoprimesBelow2000InBases2To16 table of pseudoprimes below 2000 in bases 2 to 16 TableOfSmallKnodelNumbersKnFor0N26 table of small Knödel numbers K_n for 0<n<26 TableOfSmallMultiplyPerfectNumbers table of small multiply perfect numbers TableOfSomeFundamentalUnits table of some fundamental units TableOfSubtractionUpTo12