The document provides an overview of the Mean Value Theorem and Rolle's Theorem. It discusses that the Mean Value Theorem states that for any function continuous on a closed interval, there exists a point where the slope of the tangent line equals the slope of the secant line through the endpoints. Rolle's Theorem is a special case where if a function is continuous on a closed interval and differentiable on the open interval, if the function is equal at the endpoints, the derivative at some interior point is zero. Graphical interpretations are also provided to illustrate these theorems.

1. MEAN VALUE THEOREM

2. HISTORY OF MEAN VALUE THEOREM
• Mean value theorem was first defined by Vatasseri
Parameshvara Nambudiri (a famous Indian
mathematician and astronomer), from the Kerala school
of astronomy and mathematics in India.in the modern
form, it was proved by Cauchy in 1823.
• Its special form of theorem was proved by Michel Rolle
in 1691; hence it was named as Rolle’s Theorem. This
name was first used by Moritz Wilhelm Drobisch of
Germany in 1834 and by Giusto Bellavitis of Italy in
1846.

3. MEAN VALUE THEOREM
The mean value theorem says that for any given arc between
two endpoints, there must be at least one point at which the
slope of the tangent to the arc is same as the slope of the
secant through its endpoints.
If f(x) is a function, so that f(x) is continuous on the close
interval [a, b] and also differentiable on the open interval (a, b),
then there is point c in (a, b) i.e. a < c < b such that
The mean value theorem is also known as Lagrange’s
mean value theorem or first mean value theorem.

4. GRAPHICAL INTERPRETATION
• Here the above figure shows the graph of function f(x).
• Let A = (a, f (a)) and B = (b, f (b))
• At point c where the tangent passes through the curve is (c,
f(c)).
• The slope of the tangent line is same as secant line i.e. both
are parallel to each other

5. ROLLE’S THEOREM
• It is a special case of mean value theorem .
• It says that if a function f(x) is continuous on the closed
interval [a, b] and differentiable on the open interval (a, b),
then there must be one point c which comes between a and b
so that a < c < b. If f (a) = f (b) then
fꞌ(c) =0
• I.e. the first derivative is zero or you can say that the slope of
the tangent line to the curve of function is zero.

6. GRAPHICAL INTERPRETATION
Here the graph of function y = f(x) is continuous between
x = a and x = b. there could be so many tangents but one
tangent is there which is parallel to x axis. At that point
fꞌ(c) = 0.

7. HEY FRIEND,
This was just a summary on Mean Value Theorem. For more
detailed information on this topic, please type the link given below
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www.askiitians.com/iit-jee-applications-of-derivatives/rolle-
theoram-and-lagrange-mean-value-theorem/