The mean value theorem is considered to be the most powerful tool in the world of calculus. In both differential and integral calculus, this particular theorem plays a very important role because it is capable of solving different kinds of situations very easily and the best benefit is that it will be always capable of providing the people with the complete opportunity of understanding the identical behaviour of different kinds of functions. The hypothesis and conclusion of the mean value theorem will also help in showing some of the similarities to the people related to the intermediate value theorem. This particular theorem is also abbreviated as MVT.

## As per the mean value, theorem statements suppose F into X is a particular function then it will satisfy two conditions which are explained as follows:

- The function X will be continuous in A, B
- Function X will be differentiable in A, B

Then there will also exist the number which will be B greater than C and C greater than A

This particular theorem can even be proved with the help of a comprehensive process in which the individuals need to consider the line that will pass through the points A and function of A, point B and function of B. Hence, with the utilisation of Rolle’s Theorem, the proof of this particular theorem can be perfectly undertaken without any kind of problem. Hence, to be aware of this particular concept in every aspect associated with the mean value theorem the individuals need to have a good command over the fundamental theorem of calculus which is only possible if they enrol themselves on platforms like Cuemath because there they will be taught by the experts who will be very much successful in terms of clearing every aspect of the application of such things and will make sure that students will be scoring well in the exams.

## The Corollaries of the mean value theorem are explained as follows:

- If the function of X will be zero at each point of X then the open interval will be the function of X is equal to C where C will be the constant.
- If the function of X will be equal to the function of GX then in the open interval there will be a constant C that will be the function of X is equal to the function of GX plus C.

## The application of the mean value theorem has been explained as follows:

This will be the very basic relationship between the derivative of the function and the increasing or decreasing nature of the function. It is also very much successful in terms of defining the derivative of a differential and continuous function and the following are some of the most important few results which are used in this particular theorem:

If the function F will be there then continuous interval AB and differential interval AB will also be there.

- If there will be the functions F and G then they will be continuous in the interval A, B and will be differential on the interval A, B.
- It will also be based upon the strictly increasing function in which they will be continuous in the interval A, B and differential in interval A, B.
- They will also be strictly decreasing function and the rest of the statement will be the same as above.

With the help of this particular theorem, the approximate derivative of any function will be easily available and that theorem will also be very much successful in terms of building the relationship between the slope of the tangent line and the secant line on the curve. It will also help in showing that the actual slope will be equal to the average slope at a particular point of time in the close interval and geometrically there will be two endpoints of the curve that will help in fulfilling the overall conditions very easily and efficiently.

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