Relative Maxima and Minima of functions

Maxima is the maximum point of a function and Minima is the minimum point of a function. this is the basic among these. Both points are opposite to each other. functions are nothing but the input and the output. We use certain functions as input and then find the output of that function.

The relative maxima and relative minima are used to calculate the Breakdown voltage of electric equipment, Solve problems etc.

Maxima and Minima:

Maxima is the maximum value or maximum point of a function. The word maxima are the plural of maximums. Minima is the minimum value or minimum point of a function. The word Minima is the plural of minimums.

Local Maxima and Minima:

It is the maximum value or the maximum point in a particular time interval and the values of the function near that point is less than it. It’s called Local Maxima. And similarly, Local Minima is the minimum value or the minimum point in a particular time interval and the values of the function near that point is more than it. It’s called Local minima.

More than one local Maxima and local Minima can be present in a single graph or of a single function.

Relative Maxima and Minima:

It is the relative value of local maxima and local minima that today. The maximum value or point of a function after which the function changes its directions from increasing to decreasing is the relative Maxima ( maximum amongst all) and the minimum value or point of a function after which the function changes its directions to decreasing to increasing is the relative Minima (minimum amongst all).

Methods to find Relative Maxima and Minima of a function:

The relative maxima and minima of a function can be identified by using two derivative tests; The first derivative test and the second derivative tests.

Both the tests are useful to find the Relative and Minima value or point of a function.

To find Relative Maxima and relative Minima the steps to follow are :

1. First derivative test

2. Put f'(x)=0 and find the values and critical points.

3. Put the values in the original function.

4. Second derivative test

5. Put the same values of roots (that you got when you put f'(x)=0 ) in f”(x)

6. Then you will get the  Relative maxima and minima values

7. Plot a graph( using the values)

First derivative test :

This method is also known as finding out critical points. That is the critical point of this can be obtained by deriving the function. 

At first, 

If f(x) is a function then putting the first derivative of the function to zero which is f'(x)=0 gives the critical point of the function. 

Here, x=0 is the critical point of the function.

Now to find out whether it’s a Minima or maxima we have to go to the second step which is a second derivative test.

Second derivative test:

Then we have to do the second derivative test to find out whether it’s a Minima or Maxima. Now deriving f'(x) we will get f”(x). Now we have 2 cases:

1. If f”(x) >0; is the relative Minima point.

2. If f”(x) <0; is the relative Maxima point.

3. If f”(x) =0; is called as the point of inflection or test failed.

Example:

Find the value at the critical point of the function f(x)= 2x²-4

Ans: f(x)= 2x²-4

F'(x)= 4x

To find critical points putting f'(x)=0,

4x=0

; x=0

This is the critical point of this function.

Putting x=0 in the original function we will get

f'(x)= -4

Now to find out whether it is Maxima or Minima,

F”(x) = 4

F”(x)>0; it’s a Minima.

To get value at critical point put x=0 in the original function,

F(0)= -4 

(Ans)

Uses of relative Maxima and minima: 

• Solving problems

• To calculate the price of a stock

• To calculate Humidity level for food storage

• The breakdown voltage of an electric equipment

Conclusion:

The function of relative Maxima and relative Minima is useful to calculate the maximum and minimum value of a function and plotting a graph using the value is quite easy. Just put the values in the graph and join them you will get local Maxima, local minima, relative Maxima & relative Minima.