__Mean Value Theorem:__

If a function* f* Â is defined on the closed interval [a,b] satisfying the following conditions –

i) The function *f* is continuous on the closed interval [a, b]

ii)The function *f* Â is differentiable on the open interval (a, b)

Then there exists a value Â x = Â c in such a way that

\(~~~~~~~~~~~\) f'(c) = \( \frac {f(b) – f(a)}{b-a}\)

This theorem is also known as first mean value theorem or Lagrangeâ€™s mean value theorem.

__Geometrical Interpretation of Lagrangeâ€™s Mean Value Theorem:__

In the given graph the curve y = *f*(x) Â is continuous from x = a and x = b and differentiable within the closed interval [a,b] then according to Lagrangeâ€™s mean value theorem,for any function that is continuous on [*a*,Â *b*] and differentiable on (*a*,Â *b*) there exists someÂ *c*Â in the interval (*a*,Â *b*) such that the secant joining the endpoints of the interval [*a*,Â *b*] is parallel to the tangent atÂ *c*.

\(~~~~~~~~~~\) \( f'(c) \) = \( \frac {f(b) – f(a) }{b -a} \)<

__Rolleâ€™s Theorem:__

A special case of Lagrangeâ€™s mean value theorem is Rolle â€™s Theorem which states that:

If a function *f* Â is defined in the closed interval [a,b] in such a way that it satisfies the following conditions.

i) The function *f* is continuous on the closed interval [a, b]

ii)The function *Â f* is differentiable on the open interval (a, b)

iii) Now if *f (a) = fÂ *(b)Â , then there exists at least one value of x, let us assume this value to be c, whereas Â lies between Â a and b i.e. (a < c < b ) Â in such a way that *f*‘(c) = 0 .

Precisely, if a function is continuous on the closed interval [a, b] and differentiable on the open interval (a, b) then there exists a point x = c Â in (a, b) such that f'(c) = 0

__Geometric interpretation of Rolleâ€™s Theorem:__

In the given graph, the curve y =* f*(x) Â is continuous between x =a Â and x = b and at every point within the interval it is possible to draw a tangent and ordinates corresponding to the abscissa and are equal then there exists at least one tangent to the curve which is parallel to the x-axis.

Algebraically, this theorem tells us that if *f* (x) is representing a polynomial function in x and the two roots of the equation *f*(x) = 0 are x =a and x = b, then there exists at least one root of the equation *f*‘(x) = 0 Â lying between the values.

Now I assume we are clear with what actually Rolleâ€™s Theorem is. It should also be duly noted that the converse of Rolleâ€™s theorem is not true and it is also possible that there exists more than one value of x = v for which the theorem holds good but there is a definite chance of the existence of one such value.

This is all about mean value theorem and Rolleâ€™s theorem. Explore more concepts of Differential Calculus with BYJU’S.