Module 13 : Maxima, Minima and Saddle Points, Constrained maxima and minima
Lecture 39 : Constrained maxima / minima [Section 39.2]
39.2.2
Theorem ( Lagrange multiplier theorem):
Let and be such that the the following holds:
(i)
Both the partial derivatives of and exist in and are continuous at
(ii)
and
(iii)
The function has a local extremum at , when restricted to the level curve
Then,
Proof
Since , we have
Suppose, . Then, using implicit function theorem, we can find a function such that
Hence, by chain rule, ----------(32)
Also, since has a local extremum at the point when restricted to , if we define
then has a local extremum at . Therefore, ----------(33)
It follows from the equations (32) and (33),
and hence
where
and 
and 
exist in 
and are continuous at 
and 
has a local extremum at 
, when restricted to 
the level curve 
, we have 
. Then, using implicit function theorem, we can find a function 
such that 
----------(32) 
has a local extremum at the point 
when restricted to 
, if we define 
has a local extremum at 
. Therefore, 
----------(33)
