Use Lagrange multipliers to find the optimal value of the function 𝑓(𝑥,𝑦)=𝑥2+𝑦2+𝑥𝑦 subject to the constraint 𝑥−2𝑦=7 Find the location of the optimal point, the value of 𝜆 and the value of 𝑓(𝑥,𝑦) at this point. Enter non-integer numerical values as decimals to at least 3 decimal places. Note: you must use a . and not , for a decimal point. The optimal point is located at 𝑥= Answer 1 Question 5 , 𝑦= Answer 2 Question 5 , with 𝜆= Answer 3 Question 5 , and 𝑓(𝑥,𝑦)= Answer 4 Question 5
Question
Use Lagrange multipliers to find the optimal value of the function
𝑓(𝑥,𝑦)=𝑥2+𝑦2+𝑥𝑦
subject to the constraint
𝑥−2𝑦=7
Find the location of the optimal point, the value of 𝜆 and the value of 𝑓(𝑥,𝑦) at this point. Enter non-integer numerical values as decimals to at least 3 decimal places. Note: you must use a . and not , for a decimal point.
The optimal point is located at 𝑥= Answer 1 Question 5
, 𝑦= Answer 2 Question 5
, with 𝜆= Answer 3 Question 5
, and 𝑓(𝑥,𝑦)= Answer 4 Question 5
Solution
To solve this problem, we will use the method of Lagrange multipliers. This method is used to find the local maxima and minima of a function subject to equality constraints.
Step 1: Set up the Lagrangian
The Lagrangian is the original function plus a new variable (the Lagrange multiplier, λ) times the constraint function. In this case, the Lagrangian L is:
L(x, y, λ) = x² + y² + xy - λ(x - 2y - 7)
Step 2: Take the partial derivatives
We take the partial derivatives of L with respect to x, y, and λ, and set them equal to zero:
∂L/∂x = 2x + y - λ = 0 ∂L/∂y = 2y + x - 2λ = 0 ∂L/∂λ = x - 2y - 7 = 0
Step 3: Solve the system of equations
We now have a system of three equations in three variables. Solving this system gives us the values of x, y, and λ that optimize the function subject to the constraint.
From ∂L/∂λ = 0, we have x = 2y + 7. Substituting this into ∂L/∂x = 0 gives us 2(2y + 7) + y - λ = 0, which simplifies to 5y + 14 - λ = 0.
Substituting x = 2y + 7 into ∂L/∂y = 0 gives us 2y + 2y + 7 - 2λ = 0, which simplifies to 4y + 7 - 2λ = 0.
Solving these two equations simultaneously gives us y = -1 and λ = -9. Substituting y = -1 into x = 2y + 7 gives us x = 5.
Step 4: Find the value of the function at the optimal point
Substituting x = 5 and y = -1 into the original function f(x, y) = x² + y² + xy gives us f(5, -1) = 25 + 1 - 5 = 21.
So, the optimal point is located at x = 5, y = -1, with λ = -9, and f(x, y) = 21 at this point.
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