example of tests for local extrema in Lagrange multiplier method

Let n+ and c. We want to find the local extrema of the function


subject to the condition g=0, where


The first and second orderPlanetmathPlanetmath partial derivativesMathworldPlanetmath are for all i,j{1,,n}


where δi,j is the Kroenecker-delta. Thus the necessary condition f(x)=λg(x) together with g(x)=0 gives the system of equations


By summing the first n equations and then substituting in the last we get


Thus there is only one point, where local extremum is possible. We apply the test in the parent entry to the matrix


where P is the matrix containing 1/n in all entries, and I is the identity matrixMathworldPlanetmath. P is a rank one projection. Therefore the second derivative has spectrum σ(nP-I)={n-1,-1}, where -1 has multiplicity n-1, and n-1 has multiplicity 1. Thus the second derivative of f-λg is indefinit, so it has no local extrema. However the nullspaceMathworldPlanetmath of g(x) is precisely the nullspace of P, thus the second derivative is strictly negative on the tangent space Tx(M), so the vector (c/n,,c/n) is a local maximum of f subject to g=0.

Title example of tests for local extrema in Lagrange multiplier method
Canonical name ExampleOfTestsForLocalExtremaInLagrangeMultiplierMethod
Date of creation 2013-03-22 19:12:19
Last modified on 2013-03-22 19:12:19
Owner scineram (4030)
Last modified by scineram (4030)
Numerical id 9
Author scineram (4030)
Entry type Example
Classification msc 26B12
Classification msc 49K35
Classification msc 49-00