Use Lagrange multipliers to find the maximum value of f (x, y, z) = 4xyz subject to 3x + 3y + 2z = 54.
A) f (7, 4, 9) = 1,008
B) f (0, 0, 0) = 0
C) f (9, 7, 7) = 1,764
D) f (6, 6, 9) = 1,296

Question

Quizplus · Accepted Answer

To solve this problem using Lagrange multipliers, we set up the following equations from the given function $f(x, y, z) = 4xyz$ and constraint $3x + 3y + 2z = 54$:1. $\nabla f(x, y, z) = \lambda \nabla g(x, y, z)$, where $g(x, y, z) = 3x + 3y + 2z - 54$.2. This gives us the system of equations:   - $4yz = 3\lambda$   - $4xz = 3\lambda$   - $4xy = 2\lambda$   - $3x + 3y + 2z = 54$From these equations, we can solve for $x$, $y$, and $z$ in terms of $\lambda$, and then use the constraint equation to find specific values for $x$, $y$, and $z$. However, without solving the system explicitly, we can check the given options against the constraint and the function to see which one maximizes $f(x, y, z)$.Option D, $f(6, 6, 9) = 4 \cdot 6 \cdot 6 \cdot 9 = 1,296$, satisfies the constraint $3(6) + 3(6) + 2(9) = 54$ and offers the highest value of $f(x, y, z)$ among the given options.

Use Lagrange Multipliers to Find the Maximum Value of F