Unformatted text preview: x , which is the same as saying that it must be true for any vector in the plane tangent to the level set L ( g,c ) at −→ x : in other words, −→ ∇ f ( −→ x ) must be normal to this tangent plane. But we already know that the gradient of g is normal to this tangent plane; thus the two gradient vectors must point along the same line—they must be linearly dependent! This proves (see ±igure 3.24 ) Proposition 3.6.11 (Lagrange Multipliers) . If −→ x is a local extreme point of the restriction of the function f ( −→ x ) to the level set L ( g,c ) of the function g ( −→ x ) , and c is a regular value of g . Then −→ ∇ f ( −→ x ) and −→ ∇ g ( −→ x ) must be linearly dependent: −→ ∇ f ( −→ x ) = λ −→ ∇ g ( −→ x ) (3.27) for some real number λ ....
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.
- Fall '08