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

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

asked Feb 18, 2015 in CALCULUS
reshown Feb 18, 2015

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

Step 3 :

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) :

edited Feb 23, 2015 by Thomas

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

Step 5 :

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