**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 .