20e-lecture11

# 20e-lecture11 - Lecture 11 Friday April 24th...

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Unformatted text preview: Lecture 11 - Friday April 24th [email protected] 11.1 Lagrange Multipliers Let f : R n → R be a function defined on a region R described by a functional equation or constraint g ( x ) = 0 and such that f ∈ C 1 ( R ). Let φ be a function defined by φ ( x,λ ) = f ( x ) + λg ( x ) where λ is called a Lagrange Multiplier and is to be determined later. Then amongst all points x ∈ R and λ ∈ R such that ∇ φ ( x,λ ) = 0, the extreme points of f is guaranteed to appear. That this is true follows from the theory of implicit functions, and this method is referred to as the method of Lagrange Multipliers. Example 1. Find the dimensions of the box of largest volume which can be fitted inside the sphere x 2 + y 2 + z 2 = 1. Solution. We can assume the sides of the boxes are parallel to the co-ordinate axes, so if ( x,y,z ) is a corner of the box on the sphere with x > 0 and y > 0 and z > 0, then the volume of the box is 8 xyz . By Lagrange’s Method, to find the maximum volume we have....
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20e-lecture11 - Lecture 11 Friday April 24th...

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