Question

Use Lagrange multipliers to find the indicated extrema, assuming that x and y are positive.

Answer 1

Step 1:

Method of Lagrange Multipliers :

If f  and g  satisfy the hypothesis of Lagrange's theorem, and let f  have a minimum or maximum subject to the constraint . To find the minimum or maximum of f  use these steps.

1. Simultaneously solve the equations  and by solving the following system of equations.

2. Evaluate f  at each solution point obtained in the first step. The greatest valueyields the maximum of subject to the constraint , and the least value yields the minimum of subject to the constraint .

Step 2 :

The function is .

The constraint is .

Consider

Find the gradient :

Find the gradient :

Step 3 :

Write the system of equations :

Solve the equation (1) :

Substitute equation (2).

Step 4 :

Substitute  equation (3).

Substitute  equation (3).

Substitute in the function .

The maximum value of the function is 2600.

Solution :

The maximum value of the function is 2600.