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| PAGE: 987 | SET: Exercises | PROBLEM: 9 |
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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