Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 14 - Partial Derivatives - 14.7 Maximum and Minimum Values - 14.7 Exercises - Page 1009: 46


$(4,4,4)$ and $48$

Work Step by Step

From the given statement let us consider $F(x,y,z)=x^2+y^2+z^2, G(x,y,z)=x+y+z=12$ Need to apply Lagrange Multipliers Method to determine the three positive numbers. we have $\nabla f=\lambda \nabla g$ $F(x,y,z)=x^2+y^2+z^2, G(x,y,z)=x+y+z=12$ or, $\lambda =2x$ and $\lambda =2y$ and $\lambda =2z$ or, $x=y=z=\dfrac{\lambda}{2}$ Simplify to get value for $x$. we get $x=4$ Therefore, $x=4,y=4,z=4$; Minimum value: $x^2+y^2+z^2=(4)^2+(4)^2+(4)^2=48$
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