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 c 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 x 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 D .
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 .