Lagrange multipliers

Lagrange multipliers

Lessons

Notes:

Lagrange Multipliers for 2 Variable Functions

Last section, we saw that it was a long process to calculate potential absolute max & mins on a boundary. Lagrange Multipliers help make this process easier and faster.

Suppose we have a function f(x,y)f(x,y), and we want to optimize this function when given a constraint function g(x,y)g(x,y). There are 2 steps we need to do:

  1. Solve the systems of equations:

    fx=λgxf_x = \lambda g_x
    fy=λgyf_y = \lambda g_y
    g(x,y)=0g(x,y) = 0

  2. Plug all the solutions (x,y)(x,y) into the function f(x,y)f(x,y) to identify any maximums & minimum.

Lagrange Multipliers for 3 Variable Functions

Suppose we have a function f(x,y,z)f(x,y,z), and we want to optimize this function when given a constraint function g(x,y,z)g(x,y,z). Once again, there are two steps

  1. Solve the systems of equations:

    fx=λgxf_x = \lambda g_x
    fy=λgyf_y = \lambda g_y
    fz=λgzf_z = \lambda g_z
    g(x,y,z)=0g(x,y,z) = 0

  2. Plug all the solutions (x,y,z)(x,y,z) into the function f(x,y,z)f(x,y,z) to identify any maximums & minimum.
  • Introduction
    Lagrange Multipliers Overview:
    a)
    Lagrange Multipliers for 2 Variable Functions
    • fx=λgxf_x = \lambda g_x
    • fy=λgyf_y = \lambda g_y
    • g(x,y)=0g(x,y) = 0
    • Identify any max & mins
    • Example

    b)
    Lagrange Multipliers for 3 Variable Functions
    • fx=λgxf_x = \lambda g_x
    • fy=λgyf_y = \lambda g_y
    • fz=λgzf_z = \lambda g_z
    • g(x,y)=0g(x,y) = 0
    • Identify any max & mins
    • Example