Explanation Given: The function is f ( x , y , z ) = x + y + z and two constraints are x 2 + y 2 + z 2 = 9 and 1 4 x 2 + 1 4 y 2 + 4 z 2 = 9 . Calculation: Here, the function is f ( x , y , z ) = x + y + z subjected to the constraints are x 2 + y 2 + z 2 = 9 and 1 4 x 2 + 1 4 y 2 + 4 z 2 = 9 .The constraint curves are g ( x , y , z ) = x 2 + y 2 + z 2 − 9 = 0 and h ( x , y , z ) = 1 4 x 2 + 1 4 y 2 + 4 z 2 − 9 = 0 . Here, the function is f ( x , y , z ) = x + y + z . We need to calculate ∇ f , ∇ g and ∇ h . Now, f ( x , y , z ) = x + y + z ∇ f = ( f x ( x , y , z ) , f y ( x , y , z ) , f z ( x , y , z ) ) = ( ∂ ∂ x ( x + y + z ) , ∂ ∂ y ( x + y + z ) , ∂ ∂ z ( x + y + z ) ) = ( 1 , 1 , 1 ) The constraint curve is g ( x , y , z ) = x 2 + y 2 + z 2 − 9 . Then, g ( x , y , z ) = x 2 + y 2 + z 2 − 9 ∇ g = ( g x ( x , y , z ) , g y ( x , y , z ) , g z ( x , y , z ) ) = ( ∂ ∂ x ( x 2 + y 2 + z 2 − 9 ) , ∂ ∂ y ( x 2 + y 2 + z 2 − 9 ) , ∂ ∂ y ( x 2 + y 2 + z 2 − 9 ) ) = ( 2 x , 2 y , 2 z ) The other constraint curve is h ( x , y , z ) = 1 4 x 2 + 1 4 y 2 + 4 z 2 − 9 . Then, h ( x , y , z ) = 1 4 x 2 + 1 4 y 2 + 4 z 2 − 9 ∇ h = ( h x ( x , y , z ) , h y ( x , y , z ) , h z ( x , y , z ) ) = ( ∂ ∂ x ( 1 4 x 2 + 1 4 y 2 + 4 z 2 − 9 ) , ∂ ∂ y ( 1 4 x 2 + 1 4 y 2 + 4 z 2 − 9 ) , ∂ ∂ y ( 1 4 x 2 + 1 4 y 2 + 4 z 2 − 9 ) ) = ( 1 2 x , 1 2 y , 8 z ) Then, the Lagrange equation is ∇ f = λ ∇ g + μ ∇ h ...

4th Edition

ISBN: 9781319050740

Author: Jon Rogawski, Colin Adams, Robert Franzosa

Publisher: W. H. Freeman

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Question

Chapter 14.8, Problem 44E

To determine

The maximum of $f (x, y, z) = x + y + z$ subject to the two constraints $x^{2} + y^{2} + z^{2} = 9$ and $\frac{1}{4} x^{2} + \frac{1}{4} y^{2} + 4 z^{2} = 9$ .

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Chapter 14.8, Problem 43E

Chapter 14.8, Problem 45E

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Chapter 14 Solutions

Calculus: Early Transcendentals

Ch. 14.1 - Prob. 1PQ Ch. 14.1 - Prob. 2PQ Ch. 14.1 - Prob. 3PQ Ch. 14.1 - Prob. 4PQ Ch. 14.1 - Prob. 5PQ Ch. 14.1 - Prob. 1E Ch. 14.1 - Prob. 2E Ch. 14.1 - Prob. 3E Ch. 14.1 - Prob. 4E Ch. 14.1 - Prob. 5E

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The maximum of f ( x , y , z ) = x + y + z subject to the two constraints x 2 + y 2 + z 2 = 9 and 1 4 x 2 + 1 4 y 2 + 4 z 2 = 9 .

The maximum of f(x,y,z)=x+y+z subject to the two constraints x2+y2+z2=9 and 14x2+14y2+4z2=9.

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Chapter 14 Solutions

The maximum of $f (x, y, z) = x + y + z$ subject to the two constraints $x^{2} + y^{2} + z^{2} = 9$ and $\frac{1}{4} x^{2} + \frac{1}{4} y^{2} + 4 z^{2} = 9$ .