Lagranges Method of Undetermined Multipliers
Lagrange’s Method of Undetermined Multipliers:
So far, we have considered the method of finding the extreme values of a function f (x, y, z), where these variables x, y, z are independent. Sometimes we may have to find the maximum or minimum values of a function f (x, y, z), when x, y, z are connected by some relations say φ (x, y, z) = 0. In such cases we may eliminate z from the given conditions and express the function f (x, y, z) as a function of two variables x and y and obtain the extreme values of f as before. In such cases we have an alternative method for finding the critical points called “Lagrange’s method of undetermined multipliers”.
Suppose we want to find the maximum and minimum values of the functions
Subject to the conditions;
But the necessary condition for the functions f (x, y, z) to have maximum and minimum values are f x = 0, f y = 0, f z = 0
Multiplying (4) by λ and adding it to (3), we get
This is possible only if
Solving the equations (2), (5), (6) and (7), we get the values of x, y, z and the undetermined multiplies λ. Thus, we obtain the critical points of the function f (x, y, z). But this method does not help us in identifying whether the critical points of the function gives the maximum or minimum.
To find the extreme value of the functions f (x,y,z) subject to the conditions Φ (x,y,z) = 0
1. From the auxiliary equation
2. Find the critical points of F as a functions of our variables x,y,z,λ. Solving the equations we get
Hence, The value of x,y,z thus obtained will give us the critical point (x,y,z).