Section 7.4: Maxima and Minima Using Lagrange Multipliers

Method of Lagrange Multipliers For Functions of Two Variables

Any local maxima or minima of the function z = f(x, y) subject to the constraint g(x, y) = 0 will be among those points (x0, y0) for which (x0, y0, λ0) is a solution of the system:

Fx(x, y, λ) = 0
Fy(x, y, λ) = 0
Fλ(x, y, λ) = 0

where F(x, y, λ) = f(x, y) + λg(x, y), provided that all the partial derivatives exist.

x = 40

Method of Lagrange Multipliers: Key Steps

  1. Write the problem in the form:
    Maximize (or minimize) z = f(x, y)subject tog(x, y) = 0
  2. Form the function F:
    F(x, y, λ) = f(x, y) + λg(x, y)
  3. Find the critical points of F; that is, solve the system:
    Fx(x, y, λ) = 0
    Fy(x, y, λ) = 0
    Fλ(x, y, λ) = 0
    • If (x0, y0, λ0) is the only critical point of F, we assume that (x0, y0) will always produce the solution to the problems we consider.
    • If F has more than one critical point, we evaluate z = f(x, y) at (x0, y0) for each critical point (x0, y0, λ0) of F.
    • For the problems we consider, we assume that the largest of these values is the maximum value of f(x, y), subject to the constraint g(x, y) = 0, and the smallest of these values is the minimum value of f(x, y), subject to the constraint g(x, y) = 0.

f(x, y) = 5

Method of Lagrange Multipliers For Functions of Three Variables

Any local maxima or minima of the function w = f(x, y, z) subject to the constraint g(x, y, z) = 0 will be among those points (x0, y0, z0) for which (x0, y0, z0 λ0) is a solution of the system:

Fx(x, y, z, λ) = 0
Fy(x, y, z, λ) = 0
Fz(x, y, z, λ) = 0
Fλ(x, y, z, λ) = 0

where F(x, y, z, λ) = f(x, y, z) + λg(x, y, z), provided that all the partial derivatives exist.