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Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint.

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Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint.

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

2 Answers

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Step 1:

Method of Lagrange Multipliers :

To find the minimum or maximum values of subject to the constraint .

(a). Find all values of x, y, z and such that

and .

(b). Evaluate f  at all points that results from step (a). The largest of these values is the maximum value of f,  the smallest is the minimum value of f.

Step 2 :

The function is .

The constraint is image.

Consider image

Find the gradient :

Find the gradient :

image

Step 3 :

Write the system of equations :

image

Multiply equation (1) by x :

Multiply equation (2) by y :

Multiply equation (3) by z :

Equate equation (4) and equation (5) :

Equate equation (5) and equation (6) :

image

answered Feb 23, 2015 by Thomas Apprentice
edited Feb 23, 2015 by Thomas
0 votes

Contd.............

Step 5 :

Substitute and in the constraint image.

Substitute in .

Substitute in .

The points are and .

Step 5 :

Substitute the point in the function .

Substitute the point in the function .

image

The minimum value is

The maximum value is    

Solution :

The minimum value is

The maximum value is

answered Feb 23, 2015 by Thomas Apprentice

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