 
Summary: 14.8 Lagrange Multipliers
Constrained Extrema for f(x,y,z) (with one constraint)
1
Question:
To optimize a function ( , , )f x y z
subject to a given constraint ( , , )g x y z k=
Assuming that extreme
values exist and
on the surface0g
( , , )g x y z k=
Also called finding
constrained extrema
Answer: Lagrange theorem
If ( , , )f x y z has an extrema at ( )00 0,, zx y subject to
then( , , )g x y z k= 0 00 0 0 0, ,( , ) ( , )z zf x y g x y =
is called Lagrange multiplier
Note
The solutions of this equation gives all
possible candidate for extreme points
Next
