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

7 years ago

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Answers : (4)

										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
df(x)/dx]at (x=c) ={f(b)-f(a)}/(b-a)=0

E.g in the assumed problem after applying Rolle's theorem we get

f(x)=0,it implies x*x+x+3x+3=0
or, x(x+1)+3(x+1)= 0
or,(x+1)(x+3)=0
or, x=-1, x=-3 ,it means that f(-1)=0, f(-3)=0, a = -1 and b = -3

df(x)/dx = 2x+4 =0 or, x= -2

df(x)/dx ={f(-1)-f(-3)}/{-1-(-3)}= (0-0)/2= 0

hence the value of c = -2 and -3 < -2 < -1, according to rolle's theorem a
7 years ago
										

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.


Please feel free to post as many doubts on our discussion forum as you can. If you find any question
Difficult to understand - post it here and we will get you the answer and detailed solution very quickly. We
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6 years ago
										

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.

4 years ago
										

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

4 years ago

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