MATH 2204 Lecture Notes - Lecture 21: Lagrange Multiplier
2 Dec 2018
School
Department
Course
Professor
Lagrange Multipliers
Daniel Eisert | Math-2204
Lambda in the above equation is a Lagrange multiplier. If we write the vector equation ∇𝑓 = 𝜆∇𝑔 in
terms of components, then the above equation becomes the following:
Similarly, we can find the extreme values of 𝑓(𝑥, 𝑦) subject to the constraint 𝑔(𝑥, 𝑦) = 𝑘, we look for
values of x, y, and 𝜆 to eventually solve the following three equations in three unknowns:
• Two Constraints: we can find the maximum and minimum values of a function subject to two
constraints (side conditions) of the form 𝑔(𝑥, 𝑦, 𝑧)= 𝑘 and ℎ(𝑥, 𝑦, 𝑧)= 𝑐. Geometrically, this
means that we are looking for extreme values of f when (𝑥, 𝑦, 𝑧) is restricted to lie on the curve
of intersection C of the level surfaces 𝑔(𝑥, 𝑦, 𝑧)= 𝑘 and ℎ(𝑥, 𝑦, 𝑧)= 𝑐.
14.8 Examples
14.8
