6.7
The Method of Lagrange Multipliers
Definition 35 The Lagrange function associated with the function ( ) and the equation ( ) = 0 is
( ) = ( ) + · ( ) where is the Lagrange multiplier.
Definition 36 Suppose ( ) is the Lagrange function associated with ( ) and a constraint equation. Also, suppose ( ) is such that
( ) = 0;
( ) = 0;
( ) = 0
Then ( ( )) is a constrained critical point on the graph of ( )
Remark 6 A constrained critical point is also a constrained point. So that
( ) = ( )
where ( ) = 0 is the constraint equation.
Theorem 75 Suppose ( ) is a partial differentiable function whose in-dependent variables are subject to a constraint. Excluding points on the edge of the graph, the high and low constrained points on the graph are among the constrained critical points.
Example 169 Use the method of Lagrange multipliers to find the values of and that
minimize ( ) = 2+ 2− 16 − 12 + 105 subject to the constraint 2 + = 7
