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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 off, the smallest is the minimum value of f.
The function is .
The constraint is .
Consider
Find the gradient :
Find the gradient :
Write the system of equations :
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) :
Substitute and in the constraint .
Substitute in .
Substitute in .
The points are and .
Substitute the point in the function .
Substitute the point in the function .
The minimum value is
The maximum value is
The minimum value is
The maximum value is
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