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 .

Consider

Find the gradient :

Find the gradient :

Step 3 :

Write the system of equations :

Multiply equation (1) by x :

Multiply equation (2) by y :

Multiply equation (3) by z :

Step 4 :

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 .

Step 5 :

Substitute the point in the function .

Substitute the point in the function .

The minimum value is

The maximum value is    

Solution : 

The minimum value is

The maximum value is

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 .

Consider

Find the gradient :

Find the gradient :

Step 3 :

Write the system of equations :

Multiply equation (1) by x :

Multiply equation (2) by y :

Multiply equation (3) by z :

Step 4 :

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 .

Step 5 :

Substitute the point in the function .

Substitute the point in the function .

The minimum value is

The maximum value is    

Solution :

The minimum value is

The maximum value is