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).   Locate the interior critical points by solving the equations                                        ∂f/ ∂x   =   0                                        ∂f/ ∂y   =   0   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.