MATH 20C Lecture 23

14.8 Lagrange Multipliers

Theorem

Assume and are differentiable.
If has 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 point satisfying 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 optimize subject to the constraint
The critical points can be found by solving