The Method of Lagrange Multipliers

The Method of Lagrange Multipliers:

The method says that the extreme values of a function
are to be whose variables are subject to a constraint
found on the surface g = 0 at the points where
(called a Lagrange Multiplier). for some scalar where

Notation:
and their first partial and Suppose that
derivatives are continuous.
To find the local maximum and minimum values of f subject to , , the constraint
that satisfy the equations: we find the values of x, y, z and
and
And


Example:
Find the maximum and minimum values of the function on the circle
Solution:
and
and



gives: and equation

has extreme values at the point:

at the points by calculating the value of

, we see that its maximum and minimum
are: values on the circle






Example:
Find the minimum value of the function

Subject to

Solution:
and
and





gives: and equation


has an extreme values at the points:

by calculating the value of
, we see that the minimum at the points
value of

Exercise:
Find the greatest and smallest values that the function

takes on the