##### Maxima and Minima Notes.

let's understand in an informal way:

Suppose a function f(x) is given and some boundary value is associated with it for example:

and boundary value is {0, 6}

The only questions that can be asked are:-

1. Find the points of local maxima and minima.

2. Find the maximum or minimum value of f(x).

**Critical points:-**

let us suppose a function f(x) is given and a point a is defined as a critical point if

**f'(a) =0.**

Sometimes it is also called as** stationary points.**

**Method to find Maxima and Minima:-**

let us say our function is represented as f(x)

1. first we have to find out f'(x)

2. Now find stationary points by **f'(x) =0**. you may get one or more critical points. Suppose 'a' is our critical point.

3. Now find out **f''(x)**

**Case1: If f''(a) < 0** then '**a' is called point of maxima** and f(x) has maximum value at this point.

**Case 2: if f''(a) >0** then **'a' is called the point of minima** and f(x) has a minimum value at this point.

Case 3: if f''(a) =0 then we need to derivative test.

**Derivative test.**

Informally, when we get even derivative like **f''(a) =0** then we have to apply double derivate test.

We need to find odd derivate **f'''(x)**

Case 1:** if f'''(a) = 0** ,then find further even derivate like f''''(x).

Case 2: **if f'''(a) ! = 0**, then the given function doesn't have any maxima or minima.

https://www.techtud.com/example-tbd/maxima-and-minima-example