Maxima and Minima 2 : Applications of Derivatives
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There are various applications of differentiation. In this course, we learn to apply derivatives to find the maximum and minimum values of differentiable functions in their domains. To begin with in the first section, a brief note about the need to study the topic Maxima and Minima is given.
In sections 2,3,4 the definitions and the concepts of the points of local / global /absolute maxima and minima which can be obtained by using differentiation is discussed.Also the behavior of f ‘(x) at local maxima and local minima points is discussed.
In section three, the terms stationary points, critical points and points of inflexion are taken up. In this section we also discuss about the concept of concavity, concave upward curves and concave downward curves. Also we see how the concept of concavity is applied to identify the points of inflexion.
The next section deals with various derivative tests for local maximum and local minimum. The tests discussed are the first derivative test, the second derivative test and in general the higher order derivative test. Working rule to use these tests is also included at the end of the lectures. Also downloadable supplementary material is provided under the heading "Concepts to Remember" . This covers the key concepts covered lecturewise.
Every concept is well explained with appropriate graphical figures.Every topic includes videos of examples which have been carefully selected and properly graded and solved to illustrate the concepts and techniques. Wherever possible the solutions include graphical explanations as well. At the end of the course the applications of maxima and minima under the heading 'optimization problems' have been discussed.
This topic is very important and useful for higher studies in Science, Technology and Economics in optimization problems. For example in Economics, we can tackle the problems like 1.Minimize cost production.i.e. expenses, effort etc.
2.Maximize profits, efficiency and outputs etc.
In Mensuration, we can find the solutions to the problems where we need to maximise or minimise the volumes or areas of geometric figures such as cylinder, cuboid etc.
However, we are today equipped with graphing calculators and computers to find the maximum and minimum values of functions.
But having said that it is still required to study this topic of “ Maxima and Minima” in Calculus to increase our understanding of functions and the mathematics involved.
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Section 1: About the Course  

Lecture 1  7 pages  
In this lecture we understand the need to study the topic Maxima and minima. 

Section 2: Maximum and Minimum values of a function.  
Lecture 2  08:20  
In this video you will understand when a function is said to attain a maximum value and a minimum value in its domain.


Lecture 3  11:51  
In this video you will understand local maximum value and local minimum values of a function. 

Lecture 4  06:54  
In this video you will understand global maximum value and global minimum values of a function. 

Lecture 5  07:31  


Lecture 6  09:52  
In this video you will understand the behavior of f ‘(x) at local maxima and local minima. 

Section 3: Stationary, Critical and points of Inflexion, and Concavity  
Lecture 7  09:22  
In this video you will understand the terms stationary points, critical points and points of inflexion. 

Lecture 8  10:43  
In this video you will understand the the concept of concavity and hence the terms concave upward and concave downward. Understand points of inflexion with the help of concavity. 

Section 4: Derivative Tests for Local Maximum and Local Minimum  
Lecture 9  05:52  
In this video you will understand first order derivative test to find local maximum and local minimum points and their respective values.. Also working rule to find the same. 

Lecture 10  05:26  
Lecture Description:
In this video you will understand second order derivative test to find local maxima and local minima. Also working rule to find the same. 

Lecture 11  9 pages  
In this lecture you will understand the higher order derivative test to find local maxima and local minima. Also working rule to find the same. 

Section 5: Examples on maximum and minimum  
Lecture 12  06:04  
In this video we will discuss an example to find the maximum or minimum values, if any of a given function in its domain without using derivatives. 

Lecture 13  05:24  
In this video we will discuss an example to find the maximum or minimum values, if any of a given function in its domain without using derivatives. 

Lecture 14  02:32  
In this video we will discuss an example to find the maximum or minimum values, if any of a given function in its domain without using derivatives. 

Lecture 15  03:26  
In this video we will discuss an example to find the maximum or minimum values, if any of a given function in its domain without using derivatives. 

Lecture 16  02:45  
In this video we will discuss an example to find the maximum or minimum values, if any of a given function in its domain without using derivatives 

Lecture 17  05:06  
In this video we will discuss an example to find the maximum or minimum values, if any of a given function in its domain without using derivatives. 

Lecture 18  03:17  
In this video we will discuss an example to find the maximum or minimum values, if any of a given function in its domain without using derivatives. 

Lecture 19  02:58  
In this video we will discuss an example to find the maximum or minimum values, if any of a given function in its domain without using derivatives. 

Lecture 20  02:02  
In this video we will discuss an example to find the maximum or minimum values, if any of a given function in its domain without using derivatives. 

Lecture 21  02:54  
In this video we will discuss an example to find the maximum or minimum values, if any of a given function in its domain without using derivatives. 

Section 6: Examples on First Derivative Test  
Lecture 22  07:56  
In this video we will discuss an example to find local maximum value and local minimum values of a function using the first order derivative test. 

Lecture 23  12:42  
In this video we will discuss an example to find local maximum value and local minimum values of a function using the first order derivative test. 

Lecture 24  14:16  
In this video we will discuss an example to find local maximum value and local minimum values of a function using the first order derivative test 

Lecture 25  06:41  
In this video we will discuss an example to find local maximum value and local minimum values of a function using the first order derivative test. 

Lecture 26  09:10  
In this video we will discuss an example to find local maximum value and local minimum values of a function using the first order derivative test. 

Lecture 27  06:10  
In this video we will discuss examples of some functions that do not have a local maximum or a local minimum. 

Section 7: Examples on Second Derivative Test  
Lecture 28  06:21  
In this video we will discuss an example to find local maximum value and local minimum values of a function using the second order derivative test. 

Lecture 29  09:23  
In this video we will discuss an example to find local maximum value and local minimum values of a function using the second order derivative test. 

Lecture 30  08:51  
In this video we will discuss an example to find local maximum value and local minimum values of a function using the second order derivative test. 

Lecture 31  07:09  
In this video we will discuss an example to find local maximum value and local minimum values of a function where the second order derivative test fails and hence use first derivative test to find the same. 

Lecture 32  11:38  
In this video we will discuss an example to find local maximum value and local minimum values of a function using the second order derivative test. 

Lecture 33  14:12  
In this video we will discuss an example to find local maximum value and local minimum values of a function using the second order derivative test. 

Lecture 34  09:24  
In this video we will discuss an example to find local maximum value and local minimum values of a function using the second order derivative test. 

Section 8: Examples on Absolute Maximum and Absolute Minimum in a closed interval  
Lecture 35  06:47  
An example to find the maximum (absolute maximum) or minimum (absolute minimum) values, of a given function defined in a closed interval. 

Lecture 36  07:55  
An example to find the maximum (absolute maximum) or minimum (absolute minimum) values, of a given function defined in a closed interval. 

Lecture 37  08:03  
An example to find the maximum (absolute maximum) or minimum (absolute minimum) values, of a given function defined in a closed interval. 

Lecture 38  06:42  
An example to find the maximum (absolute maximum) or minimum (absolute minimum) values, of a given function defined in a closed interval. 

Section 9: Optimization problems  
Lecture 39  9 pages  
A brief note on optimization problems will be given. 

Lecture 40  05:25  
To show that of all the rectangles with a given perimeter, the square has the largest area. 

Lecture 41 
Example2_ Square has the smallest perimeter

06:42  
Lecture 42  08:23  
Solution of the problem: "A square sheet of metal is to be made into an open box by cutting smaller squares from its corners and folding up the flaps to form the sides. What should be the side of the square to be cut off so that the volume of the box is maximum?" 

Lecture 43  10:09  
Solution of the problem: To show that of all the rectangles inscribed in a given fixed circle,the square has the maximum area. 

Lecture 44  08:01  
Solution of the problem: To show that the height of a closed cylinder of given surface and maximum volume is equal to the diameter of its base. 

Lecture 45  10:54  
Solution of the problem: A wire of length 40 metres is to be cut into two pieces. One of the two pieces is to be made into a square and the other into a circle. What should be the lengths of the two pieces so that the combined area of the square and the circle is minimum? 

Lecture 46  08:05  
Solution to the problem: Find two positive numbers x and y such that their sum is 35 and the product x^{2 }y^{5} is maximum. 

Lecture 47  07:03  
Solution to the problem: A closed right circular cylinder has a volume of 2156 cubic units.What should be the radius of its base so that its total surface area may be minimum? 

Lecture 48  09:37  
Solution to the problem: Prove that the area of a rightangled triangle of a given hypotenuse is maximum when the triangle is isosceles. 

Section 10: Course wrapup  
Lecture 49  5 pages  
This gives a summary of what was taught in this course. 
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