# 14.8 Notes - Section 14.8 The Lagrange Multipliers...

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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

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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
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14.8 Notes - Section 14.8 The Lagrange Multipliers...

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