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

1. Solve the systems of equations:

$f_x = \lambda g_x$
$f_y = \lambda g_y$
$g(x,y) = 0$

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

Lagrange Multipliers for 3 Variable Functions

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

1. Solve the systems of equations:

$f_x = \lambda g_x$
$f_y = \lambda g_y$
$f_z = \lambda g_z$
$g(x,y,z) = 0$

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

b)
Lagrange Multipliers for 3 Variable Functions
• $f_x = \lambda g_x$
• $f_y = \lambda g_y$
• $f_z = \lambda g_z$
• $g(x,y) = 0$
• Identify any max & mins
• Example