Module 13 : Maxima, Minima and Saddle Points, Constrained maxima and minima
Lecture 39 : Constrained maxima / minima [Section 39.2]
Mathematically, the problem is to find the absolute maximum/minimum of a function
In case we can solve for one of the variables in terms of the other, the problem can be reduced to a problem of one variable. But, often this is difficult. A method to handle such problems, without having to solve the constraint equation and giving preference to one of the variables. This method is based on the following theorem:
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
for one of the variables in terms of the other, the problem can be reduced to a problem of one variable. But, often this is difficult. A method to handle such problems, without having to solve the constraint equation and giving preference to one of the variables. This method is based on the following theorem: 
and 
and 
exist in 
and are continuous at 
and 
has a local extremum at 
, when restricted to 
the level curve 
