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

```

7 years ago

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```										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 are all IITians and here to help you in your IIT JEE preparation.
All the best  !!!

Regards,
MOHIT

```
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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