Maxima and Minima with Side Conditions Try to find the maxima and minima of the function f(r,y,z)subject to a side condition g(,y,)=0. Main Meny C 007e
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Example 1.Maximize the product zy subject to the side condition x+y-1=0. Main Meny C
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Example 2.Find the maximum volume of a rectangular solid given that the sum of the lengths of its edges is 12a. Main Menu cmew59时
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Maxima and Minima with Side Conditions The Method of Lagrange Let f be a function of two or three variables which is continuously differentiable on some open set U. We take C:r=r(t),teI to be a curve that lies entirely in Uand has at each point a nonzero tangent vector r'(t). if xo maximizes (or minimizes)f(x)on C. then Vf(xo)is perpendicular to C at xo. Main Meny
Main Menu Maxima and Minima with Side Conditions The Method of Lagrange Let f be a function of two or three variables which is continuously differentiable on some open set U. We take C : r = r (t), t I to be a curve that lies entirely in U and has at each point a nonzero tangent vector r'(t)
if xo maximizes (or minimizes)f(x)subject to the side condition g(x)=0,then Vf(xo)and Vg(xo)are parallel.Thus,if Vg(xo)≠0，then there exists a scalar入such that Vf(xo)=入Vg(xo). Such a scalar A has come to be called a Lagrange multiplier g川财 level curves: f(.y) C:g(x.y)=0 Figure 16.7.3 Main Meny C
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Such a scalar λ has come to be called a Lagrange multiplier
Example 3.Maximize and minimize f(x,y)=xy on the unit circle 22+y2 1. [Main Menu☐ 5
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Example 4.Find the minimum value taken on the function f(,y)=x2+(y-2)2 on the hyperbola 22-y2 1. Main Meny C 007e
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Example 5.A rectangular box without a top is to have a volume of 12 cubic feet.Find the dimensions of the box that will have minimum surface area. Main Meny C
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Example 6.Show that,of all the triangles inscribed in a circle of radius R,the equilateral triangle has the largest perimeter. [Main Menu☐ cme5时
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Lagrange condition asserted that vf =Avg at the maximum or minimum.This condition is equivalent to vf x vg =0. Example 7.Maximize and minimize f(x,y)=xy on the unit circle 22+2=1. Main Meny C 007e
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