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Sunday, 22 December 2013

Maxima and Minima

 

 Greatest  and Least Value :-

In this section we shall be concerned with the application of Calculus to determining the values of a function which are greatest or least in their immediate neighborhood technically known as  Maximum and Minimum Values .

It will be assumed that    f (x)   possesses continuous derivatives of every order that come in equation .

Maximum Value of a Function :-

Let     be any interior points of the interval of definition of a function   f (x)    , if it is the greatest of all its value for values of   x   lying in some neighborhood of   c   . To b more definite and to avoid the vague words   "Some Neighborhood" we say that    f (c)    is a maximum value of function if there exists some interval     ( c - D , c + D )       around     c   such that   .

                   f (c) > f (x) 

for all values of     other then      c   lying in the interval

So that ,      f (c)    is  maximum value of      f (x)     if .

f (c) > f ( c+ h )     i.e.       f ( c + h ) - f (c) < 0 

where value of    h   lying between   -D   and   D  .

Minimum Value of a Function :-

f (c)   is said to be a minimum value of    f (x)    , if it is the least of all its values for values of   x    lying in some neighborhood of     c    .

This is equivalent to saying that     f (c)    is a minimum value of    f (x)   ,if there exist a positive     D     such that ;

f (c) < f ( c + h ) ,   i.e.      f ( c + h ) - f (c) > 0

for value of    h    lying between    -D   and   .

For values of   h     sufficiently small in numerical value  .

The term Extreme Value is used both for a maximum as well as for a minimum value , so that    f (c)   is an extreme value if    f ( c+ h ) - f (c)     keeps on invariable sign for value of     h    sufficiently small numerically .