Optimization involving Functions of more than one Independent Variable                                          

 

1.

 

 

In a Nut Shell:  Concepts and strategies for the optimization of a function with

one independent variable, x, extend to functions with several independent variables

. 

 

 

 

2.

 

 

Recall that for a function of one independent variable the maximum or minimum of

the function, f(x), was determined by the condition (within the domain of x) by:

 

             f ’(x)  =  df/dx  =  0

 

Of course you needed to check the value of the function, f(x) at each end of its domain.

 

 

 

 

 

3.

 

Now consider a continuous function,  f(x,y), which has the two independent variables

x  and  y.  If  f(a,b) is either the absolute maximum or the absolute minimum

value of  f(x,y)  on a region  R, then the point  (a, b) is either

 

1.       An interior point of  R at which  ∂f/ ∂x   =   ∂f/ ∂y   =   0

2.       An interior pont of  R  where not both partial derivatives exist, or

3.       A point on the boundary of R.

 

 

 

 

 

 

 

 

3.

 

 

 

 

 

Steps to locate maxima and minima of f(x,y).

 

  1. Locate the interior critical points by solving the equations

 

                                     ∂f/ ∂x   =   0

 

                                     ∂f/ ∂y   =   0

 

  1. Find possible extreme values on the boundary.

 

      3.  Compare values of  f(x,y) from those found in steps 1 and 2.

 

 

Click here for an example.

 


Copyright © 2011 Richard C. Coddington

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