Question

1. Questions 1 relate to the following constrained optimization problem: maximize z(x,y)=4x^2-2xy+6y^2 subject to x+y=72. subject to . A) What is the optimal value of X?

B) What is the optimal value of Y?

C) What is the maximized value of Z?

Answer 1

To solve the maximization problem we use the Lagrangian method:

Since our equation is z(x,y)=4x²-2xy+6y² subject to the constraint x+y=72, therefore the Lagrangian will be

Differentiating this with respect to x,

(consider this as eq 1)

Differentiating the Lagrangian with respect to y now,

(consider this as eq 2)

Equating eq 1 with eq 2 we get,

using the above relationship and putting it in the constraint eq x+y=72, we get

and

A) The optimal value of x is 42.

B) The optimal value of y is 30.

C)

Solcing z for the maximum value of z we get,

z=4x²-2xy+6y²

i.e. z=4(42)²-2(42)(30)+6(30)²

z = 9936

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