Maximum values of a function

The maximum value of a function is also known as maxima. In a function, there are two points which are maximum and minimum for that given set of ranges. The maximum and minimum of a function, combined, are known as extremum. The maximum value of a function is the highest point in a curve. There are 2 types of maximum functions: local maxima and absolute maxima. In this maximum value of function study material, we will solve some examples so that we can gain a better understanding of the concept. 

Maximum Value of a Function with Examples

The maximum value of a function is the highest point in the curve. There can be multiple numbers of the maximum value for a function. We can even make out the highest point without looking at the graph. Maxima will be the highest point on the curve within the range and minima will be the lowest point in the range. 

Let us look at an example:

First equation: y=x³−3x²+6 

Firstly, differentiate the above equation and obtain the first derivative

Second equation: dy/dx=3x²−6x

Secondly, differentiate this second equation and obtain the second derivative

Third equation: d²y/dx²=6x−6.

Now, put dy/dx=0 for the critical values

3x²−6x= 0 

3x (x – 2 ) = 0

x = 0, 2

The critical values are 0 and 2 

Now, substitute x = 0 in the third equation 

d²y/dx²=6(0)−6

d²y/dx²= -6 < 0 

The sign for the second derivative is negative. So, the point of maxima is x=0 

Next, substitute x = 0 in the first equation to obtain the maximum value of the function 

Y = 0 – 0+ 6 

Y = 6 

Types of Maximum Function  

There are two types of maxima functions that exist 

• Local maxima 
• Absolute maxima

Local Maxima 

The local maxima of a function arise at a particular level. Local maxima are the value of a function at a point in a certain interval for which the value of the function near that point will always be less than the value of the function at that particular point. 

Absolute Maxima 

The highest point in a function with the whole domain is known as absolute maxima. There can only be one absolute maximum of a function in the entire domain. Absolute maxima can also be called global maxima.  

How to find the Maximum Value of a Function?

We can find the maximum value by using the first-order derivative and second-order derivative tests. This method is the fastest to calculate derivatives. Let’s look at them: 

First-order Derivative Test

In this test, we get the slope of the function. Near the maximum point, the slope increases as we go towards it and becomes zero. Similarly, near the minimum point, the slope decreases as we go towards it and then becomes zero. We use this to find whether the point is maxima or minima.

For instance, we have a function f, which is continuous till the critical point. Then the interval l and f’(c) is 0. We check the value of f’(x) from the left point to the curve and then from right to the curve and check the nature of f’(x), then the given point will be:

Local maxima: If f’(x) changes sign from positive to negative, when x increases to the point c, then f(c) gives the maximum value of the function in that range.  

Local minima: If f’(x) changes its sign from negative to positive, when x increases to the point c, then f(c) gives the minimum value of the function in that range. 

Point of inflection: If f’(x) doesn’t change, when x increases to the point c, then the point c is neither the maximum or minimum value for that function. 

Second-order Derivative Test 

In this test, initially, we find the first-order derivative of the function. If the value of the slope is equivalent to 0 at the critical point, then we find the second derivative of the function. The second-order derivative function exists in the following range:

Local maxima: f” (c )< 0

Local minima: f’’ (c) >0 

The test fails if f” (c ) is equivalent to 0. 

Important Applications of Functions

The important applications of functions used in real life are listed below:

1. Functions are used in the mathematical building blocks meant for designs.
2. These are utilised in machines that predict disasters.
3. They also help in curing diseases.
4. Functions are further used to understand the world economy.
5. They help in the floating of aeroplanes.

Conclusion 

Functions are an integral part of mathematics. These functions can be used in our daily lives and are applied in various applications. A function doesn’t need to have a maximum or a minimum value. You can identify whether the function has a maximum or minimum value by looking at the graph. It is necessary to find a critical point for finding the maximum value of a function. Hope the above study material notes on the maximum value of a function is valuable for cracking IIT-JEE exams with ease.