14.8 Notes - Section 14.8 The Lagrange Multipliers...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
Section 14.8 The Lagrange Multipliers Shubhodeep Mukherji 1 The Method of Lagrange Multipliers The method says that extrememe values of a function f ( x,y,z ) whose variables are subject to a constraint g ( x,y,z ) = 0 are to be found on the surface g = 0 at the points where f = λ g for some scalar λ , called a LagrangeMultiplier . 1.1 The Orthogonal Gradient Theorem 1.1.1 The Theorem Suppose that f ( x,y,z ) is differentiable in a region whose interior contains a smooth curve C : r ( t ) = g ( t ) i + h ( t ) j + k ( t ) k If P 0 is a point C where f has a local maximum or minimum relative to its values on C , then f is orthogonal to C at P 0 . 1.1.2 The Proof We show tha t f is orthogonal to the curve’s velocity vector at P 0 . The values of f on C are given by the composite f ( g ( t ) ,h ( t ) ,k ( t )), whose derivative with respect to t is df dt = ∂f ∂x dg dt + ∂f ∂y dh dt + ∂f ∂z dk dt = ( f )( v ) . At any point P 0 where f has a local max or min relative to its values on the curve, df dt = 0, so ( f )( v ) = 0 1
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
By dropping the z - terms , we obtain a similar result for functions of two variables. 1.1.3 The Corollary At the points on a smooth curve r ( t ) = g ( t ) i + h ( t ) j + k ( t ) k where a differentiable function f ( x,y ) takes on its local max and min relative
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 5

14.8 Notes - Section 14.8 The Lagrange Multipliers...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online