Module 13 : Maxima, Minima and Saddle Points, Constrained maxima and minima
Lecture 39 : Constrained maxima / minima [Section 39.2]
have no solutions for For gives have But then
Thus, the equation
Hence, the condition can not be dropped in theorem 39.2.1. However, is the distance between origin and a point on the surface Geometrically it is obvious that the minimum of is and this is attained at .
39.2.5
Constrained extremum for three Variables:
There is a result analogous to the two variable, to solve the problem of constrained maxima / minima for functions of three variables. We solve the equations
in the unknowns and at which and compare the values of at these points to locate the constrained maxima/minima of
39.2.6
Examples:
Let us find the points on the surface closest to the origin. This is same as minimizing the function
For 
gives have 
But then 
can not be dropped in theorem 39.2.1. However, 
is the distance between origin and a point on the surface 
Geometrically it is obvious that the minimum of 
is 
and this is attained at 
. 
and 
at which 
and compare the values of 
at these points to locate the constrained maxima/minima of 
closest to the origin. This is same as minimizing the function 
