$\"\"$

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Method of Lagrange Multipliers :

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

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

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$\"\"$ and $\"\"$ .

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(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.

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$\"\"$

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The function is $\"\"$ .

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The constraint is $\"\"$ .

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Consider $\"\"$

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Find the gradient $\"\"$ :

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$\"\"$

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Find the gradient $\"\"$ :

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$\"\"$

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$\"\"$

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Write the system of equations :

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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Multiply equation (1) by x :

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$\"\"$

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Multiply equation (2) by y :

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$\"\"$

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Multiply equation (3) by z :

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$\"\"$

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$\"\"$

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Equate equation (4) and equation (5) :

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$\"\"$

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Equate equation (5) and equation (6) :

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$\"\"$

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$\"\"$

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Substitute $\"\"$ and $\"\"$ in the constraint $\"\"$ .

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$\"\"$

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$

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The points are $\"\"$ and $\"\"$ .

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$\"\"$

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Substitute the point $\"\"$ in the function $\"\"$ .

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$\"\"$

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Substitute the point $\"\"$ in the function $\"\"$ .

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$\"\"$

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The minimum value is $\"\"$

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The maximum value is $\"\"$ \ \

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$\"\"$

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The minimum value is $\"\"$

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The maximum value is $\"\"$