STAT312 Lecture : Extrema, Lagrange multipliers.pdf
5 Apr 2014
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Extrema of: there is a multi- variate version of taylor s theorem, by which we have the expansion. 2 v0h ( )v for some stationary point rectional derivative: then: between x0 and x0 + v. let x0 be a (x0) = 0 so that the di- 0 (x0)v is 0 in any direction: if h ( ) x0 then x0 furnishes a local minimum of. 0 for in a neighbourhood of (x0) in this neighbourhood (i. e. (x0 + v) for su ciently small v 6= 0): if h ( ) x0 then x0 furnishes a local maximum of. 0 for in a neighbourhood of (x0) for su ciently small v 6= 0. (x0+v: if neither (1) nor (2) holds then (x0 + v) (x0) changes sign as v varies; we say that x0 is a saddlepoint. Often we seek extrema of multivariate functions, subject to certain side conditions ( constraints ).
