Steps to determine points of local maxima and local minima

At point of local maxima and local minima the slope of tangent drawn to the curve is zero. i.e. if  is either point of local maximum or local minimum of function  then (a) = 0 . So to obtain points of local maxima and minima, find (x) and equate it to zero. Then to decide between local maxima and minima we can use any one of the following methods:

Method - 1 (First Derivative Test)

1.     Find the value(s) of x where (x) vanishes.

Let  be one of these values.

2.     Local Maximum

is a point of local maximum if :

(x) > 0 at every point close to and to the left of  and (x) < 0

at every point close to and to the right of .

OR

> 0 and  > 0, where is a small +ve number

3.     Local Minima

is a point of local minimum if :

(x) < 0 at every point close to and to the left of  and (x) > 0 at

every point close to and to the right of .

OR

< 0 and  > 0, where is a small +ve number

Method - 2 (Second Derivative Test)

1.     Find the values of x where (x) vanishes.

Let  be one of these values.

2.     Local Maximum

is a point of local maximum if () < 0.

3.     Local Minima

is a point of local minimum if () > 0.