Problems of Maxima and Minima

The application of derivatives has a great scope in engineering, physics, and our daily lives. Derivatives are widely used in mathematics to determine the rate of change of various quantities. They are also useful in determining the approximate values, equation of tangent and normal, and the maximum (maxima) and minimum (minima) values of any given expression. The derivatives help in finding the slope of a curve and inflection point. An example of the significance of derivatives in engineering is: If we know the function of the speed of the train, then we can determine the maximum speed of the train. The maximum speed determination will help us choose the best and strongest material that can withstand the pressure of this high-speed train. Similarly, we can do it for other vehicles too. Let us briefly understand the Application of Derivatives in the Problems of Maxima and Minima.

Maxima and minima

To understand the application of Derivatives in the Problems of Maxima and Minima, we must get to know about maxima and minima and their types first.

Maxima and minima of any function are the maximum and minimum values of that function respectively within a given set of ranges. Simply put, maxima will be the highest peak in a given curve, whereas minima will be the lowest. The combination of these maxima and minima of a function is the extrema. 

Types of maxima and minima

There are two basic types of maxima and minima in calculus.

1)  Local maxima and minima: A function’s local maxima and minima arise at certain intervals. For local maxima, the value of a function at a certain period at a point in which the values of the function at that point are always greater than the values of the function near that point. 

On the other hand, the local minima of a function at a certain period at a point in which the values of the function at that point is always lesser than the values of function near that point.

2)  Absolute maxima and minima: The absolute maxima of a function is the highest point of function within the entire domain. On the other hand, the absolute minima of the function are the lowest point of function within the whole domain. They are also referred to as global maxima and minima and can only be one in the entire function of the domain.

How to find maxima and minima?

Assume f(x) be a continuous function. Then if,

1. The first derivative test

The postulates for the first derivative test for maxima and minima are as follows:

• f’(x) will change to negative from positive, then the point c (point of happening) will be the local maxima.

• f’(x) will change to positive from negative, then the point c (point of happening) will be the local minima.

• However, if it neither changes to positive nor negative, it is the inflection point.

2. Second derivative test

In the second derivative test, we need to find the values for x, when f’(x)=0. After finding the values, substitute these values for x in f”(x).

• If f”(x) >0, then it is the point for minima.

• If f”(x) < 0, we can conclude that it is the point for maxima.

• However, f”(x) =0, then we have to do the first derivative test to check whether it is local maxima, minima, or point of inflection.

Solved Question on Application of Derivatives in Problems of Maxima and Minima

Q- Suppose, f(2) = 15, and f’(2) = 10, find the value of f(2.1) using the application of derivatives.

Solution:

According to the question: 

f(2) = 15, and f'(2) = 10

To get: f(2.1)

By the approximation formula:

L(x) = f(a) + f'(a)(x−a)

L(x) = 15 + 10 (2.1-2)

L(x) = 15 + 10(0.1)

L(x) = 16

Answer: The value of f (2.1) is 16.

Conclusion 

The application of derivatives in problems of maxima and minima can be helpful in determining the maximum (maxima) and minimum (minima) values of any function. Derivatives are widely used in mathematics to determine the rate of change of various quantities. They are also useful in determining the approximate values, equation of tangent, and normal of any given expression. The derivatives help in finding the slope of a curve and inflection point. In a given curve, maxima will be the highest peak, whereas minima will be the lowest peak. The combination of these maxima and minima of a function is the extrema. There are two basic types of maxima and minima in calculus: local maxima and minima and absolute maxima and minima.