Use Lagrange multipliers to find the maximum value of f (x,y,z)= 6xyz subject to 2x + 5y + 2z = 60.
A)f (10,4,10)= 2,400
B)f (0,0,0)= 0
C)f (11,2,10)= 1,320
D)f (13,5,8)= 3,120

Question

Quizplus · Accepted Answer

To find the maximum value of $f(x,y,z) = 6xyz$ subject to the constraint $2x + 5y + 2z = 60$, we use Lagrange multipliers. Setting up the system of equations with $\nabla f = \lambda \nabla g$, where $g(x,y,z) = 2x + 5y + 2z - 60$, we solve for $x$, $y$, and $z$ under the constraint. The solution that maximizes $f(x,y,z)$ under the given constraint is $x=10$, $y=4$, $z=10$, leading to $f(10,4,10) = 6 \times 10 \times 4 \times 10 = 2400$.

Use Lagrange Multipliers to Find the Maximum Value of F