14.8 Lagrange Multipliers
Theorem
Assume
and are differentiable.
Ifhas a local max or min on the constraint curve at and if then there is a scalar s.t.
is called the Lagrange multiplier - The equation is called Lagrange condition/equations.
Explicitly,
Critical Point, Critical Value
Constraint Condition
A pointsatisfying the above equations is called a critical point for the optimization problem with a constraint and is called a critical value.
Remark
If the constraint curve
is not bounded, may not have a global max or min.
Lagrange Multiplier in three variables
The method of Lagrange multiplier also works in three variables.
We want to optimizesubject to the constraint
The critical points can be found by solving