Can some one explain that what is Rolle' s Theorem ??????????????????

It is the special case of Lagranges Mean Value Theorem( LMVT ),let consider a function, f(x)=x*x+4x+3 and it gives a finite value in the limit(a,b). Then their must be a point c,such that a

Dear Rahul,

Let f be continuous on a closed interval [a, b] and differentiable on the open interval (a, b). If f(a) = f(b), then there is at least one point c in (a, b) where f '(c) = 0.

It just says that between any two points where the graph of the differentiable function f (x) cuts the x-axis there must be a point where f'(x) = 0.

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the statement/definition of ROLLEs Theorem is as follows:-

If y=f(x) is

          (i)continuous function in[a,b]

          (ii)Derivable function in (a,b)

          (iii)f(a)=f(b)

Then, THERE EXISTS at least one Real number ''c''; cε(a,b) such that f''(c)=o

we have to observe that,

NOTE:1  The converse of the Rolle''s theorem,need not be true.

that means,  If f''(c)=o,then

                                    (i)f(x) need not be continuous on [a,b]

                                    (ii)f(x) need not be derivable on (a,b)

                                    (iii)f(a)=f(b) need not be true

NOTE:2   The geometrical interpretation of ROLLEs is there exists a tangent at x=c which is parallel to x-axis.

Dear Rahul,

Here is the  Rolle''s theorem,

the statement/definition of ROLLEs Theorem is as follows:-

If y=f(x) is

          (i)continuous function in[a,b]

          (ii)Derivable function in (a,b)

          (iii)f(a)=f(b)

Then, THERE EXISTS at least one Real number ''c''; cε(a,b) such that f''(c)=o

we have to observe that,

NOTE:1  The converse of the Rolle''s theorem,need not be true.

that means,  If f''(c)=o,then

                                    (i)f(x) need not be continuous on [a,b]

                                    (ii)f(x) need not be derivable on (a,b)

                                    (iii)f(a)=f(b) need not be true

NOTE:2   The geometrical interpretation of ROLLEs is there exists a tangent at x=c which is parallel to x-axis.

Thanking you

Suchandar