Once I set up my lambda with either gradient I struggle with finding what point I should use in order to get my max value. If you could please show how you get your answer that would be very helpful. Thank you. **Question 8**Maximize $4x + 2y + 2z$ on the sphere $x^2 + y^2 + z^2 = 19$.Options:a) There is no maximum.b) The maximum is $\frac{19\sqrt{14}}{18}$.c) The maximum is $\frac{2\sqrt{14}}{3}$.d) The maximum is $-2\sqrt{114}$.e) The maximum is $2\sqrt{114}$.f) None of these.**Solution Steps:**1. Defined function: $ f(x, y, z) = 4x + 2y + 2z $2. Gradient of $ f(x, y, z) = (4, 2, 2) $Steps involve using Lagrange multipliers:- Define $ g(x, y, z) = x^2 + y^2 + z^2 - 19 $- Gradient of $ g(x, y, z) = (2x, 2y, 2z) $- Setting $\nabla f = \lambda \nabla g$ leads to a system of equations: - $ 4 = 2\lambda x $ - $ 2 = 2\lambda y $ - $ 2 = 2\lambda z $ - Sphere constraint: $ x^2 + y^2 + z^2 = 19 $Solving the equations:- From $ 2\lambda x = 4 $ and $ 2\lambda y = 2 $: - $ x = 2/\lambda $ - $ y = 1/\lambda $ - $ z = 1/\lambda $- Substitute into the sphere equation to solve for $ \lambda $.This problem involves using calculus and linear algebra concepts to find the maximum value of a linear function constrained on a sphere, illustrating applications of optimization in multivariable calculus.

Answered: Maximize 4x+ 2 y+ 2 z

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Once I set up my lambda with either gradient I struggle with finding what point I should use in order to get my max value. If you could please show how you get your answer that would be very helpful. Thank you.

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$We have to maximize f (x, y, z) = 4 x + 2 y + 2 z on the sphere x^{2} + y^{2} + z^{2} = 19 . Take g (x, y, z) = x^{2} + y^{2} + z^{2} - 19 = 0 Thus, given problem is equivalent to Max f (x, y, z) = 4 x + 2 y + 2 z Subject to, g (x, y, z) = x^{2} + y^{2} + z^{2} - 19 = 0$

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