Use Lagrange multipliers to maximize the function subject to the following constraint: Assume that x,y,and z are positive.

A) 110
D) 10
E) no absolute maximum

Question

Use Lagrange multipliers to maximize the function  subject to the following constraint:  ​ Assume that x,y,and z are positive.
​
A) 110 
D) 10 
E) no absolute maximum

Quizplus · Accepted Answer

We need to find the maximum value of the function 11ea8970_6a91_bdd5_861b_0d9f8b893574_TB1301_11 subject to the constraint 11ea8970_6a91_bdd6_861b_dd1da20e7af2_TB1301_11.

Let's define the Lagrangian function L as:

L(x, y, z, λ) = 11ea8970_6a91_bdd5_861b_0d9f8b893574_TB1301_11 + λ(11ea8970_6a91_bdd6_861b_dd1da20e7af2_TB1301_11)

where λ is the Lagrange multiplier.

Now, we need to find the critical points of L. To do this, we need to find the partial derivatives of L with respect to x, y, z, and λ, and set them equal to zero:

∂L/∂x = 11 - λ = 0
∂L/∂y = 8 - λ = 0
∂L/∂z = 3 - λ = 0
∂L/∂λ = 11x + 8y + 3z - 100 = 0

Solving the first three equations, we get λ = 11, λ = 8, and λ = 3, respectively. However, we know that x, y, and z are positive, so only the solution λ = 3 is valid.

Substituting λ = 3 into the fourth equation, we get 11x + 8y + 3z = 100. This equation, along with the constraint 11ea8970_6a91_bdd6_861b_dd1da20e7af2_TB1301_11, forms a system of equations that we can solve to find the values of x, y, and z:

11x + 8y + 3z = 100
xyz = 10

Using substitution method, we get x = 2, y = 2, and z = 5/3.

Now, we can plug these values into the original function to get the maximum value:

11(2) + 8(2) + 3(5/3) = 22 + 16 + 5 = 43

Therefore, the maximum value of the function subject to the given constraint is 43, and the answer is choice (C).

Use Lagrange Multipliers to Maximize the Function Subject to the Following