Use Lagrange Multipliers to solve the given problem. Find the maximum value of subject to .

Question

Use Lagrange Multipliers to solve the given problem. Find the maximum value of  subject to  .

Quizplus · Accepted Answer

We need to maximize the function 11ebfe6a_2d00_8792_96cc_f56e2bec1e06_TBX8650_11 subject to the constraint 11ebfe6a_2d00_8793_96cc_1f845091f79e_TBX8650_11. Using Lagrange Multipliers method, we form the Lagrangian function:

L(x,y,λ) = 11ebfe6a_2d00_8792_96cc_f56e2bec1e06_TBX8650_11 - λ(11ebfe6a_2d00_8793_96cc_1f845091f79e_TBX8650_11)

Now, we find the partial derivatives of L with respect to x, y and λ and equate them to 0:

∂L/∂x = 2x = λ*2y = ∂L/∂y
∂L/∂λ = 11ebfe6a_2d00_8793_96cc_1f845091f79e_TBX8650_11 - 11ebfe6a_2d00_8792_96cc_f56e2bec1e06_TBX8650_11*y = 0

Solving these equations simultaneously, we get:

x = y
11ebfe6a_2d00_8793_96cc_1f845091f79e_TBX8650_11 - 11ebfe6a_2d00_8792_96cc_f56e2bec1e06_TBX8650_11*y = 0

Substituting y = x in the second equation, we get:

y = 11ebfe6a_2d00_8792_96cc_f56e2bec1e06_TBX8650_11 /11ebfe6a_2d00_8793_96cc_1f845091f79e_TBX8650_11

Substituting this value of y in the expression for x=y, we get:

x = 11ebfe6a_2d00_8792_96cc_f56e2bec1e06_TBX8650_11 /11ebfe6a_2d00_8793_96cc_1f845091f79e_TBX8650_11

Therefore, the maximum value of the function is:

11ebfe6a_2d00_8792_96cc_f56e2bec1e06_TBX8650_11*(11ebfe6a_2d00_8792_96cc_f56e2bec1e06_TBX8650_11 /11ebfe6a_2d00_8793_96cc_1f845091f79e_TBX8650_11 )^2

Simplifying this expression, we get:

(11ebfe6a_2d00_8792_96cc_f56e2bec1e06_TBX8650_11)^3 /(11ebfe6a_2d00_8793_96cc_1f845091f79e_TBX8650_11)^2

Looking at the answer options, we can see that choice A matches with our answer.

Use Lagrange Multipliers to Solve the Given Problem