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

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

asked Feb 18, 2015 in CALCULUS by anonymous
reshown Feb 18, 2015 by goushi

1 Answer

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Step 1:

Method of Lagrange Multipliers :

If f  and 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 image

Find the gradient image :

image

Find the gradient image :

image

Step 3 :

Write the system of equations :

image

image

image

Solve the equation (1) :

image

Substitute image equation (2).

image

Step 4 :

Substitute image equation (3).

image

Substitute image equation (3).

image

Substitute image in the function .

image

The maximum value of the function image is 2600.

Solution :

The maximum value of the function image is 2600.

answered Feb 23, 2015 by Thomas Apprentice

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