# Rolle theorem

1. Nov 16, 2008

### haya

1. The problem statement, all variables and given/known data

the rolle theorem said :
suppose
(i) F(X) is continous on colse interval (a,b)

(ii) F(X) is differentiable on open interval ( a,b)

(iii) F(a)=F(b)

then there is c on (a,b) such that F`(c)=0

the question is give an example :

1- satisfied (i) (ii) and not satisfy (i) and explain why c is not on the interval (a,b)?

2-satisfied (i) (iii) and not satisfy (ii) and explain why c is not on the interval (a,b)?

3-satisfied (ii) (iii) and not satisfy (i) and explain why c is not on the interval (a,b)?

I got answers each 1 and 2 but on 3 how can I got it
because if the function discontinous then so it is non diffrentiable on (a,b).

2. Nov 16, 2008

### HallsofIvy

That's not a closed interval. You mean [a, b].

But, as I pointed out above that is NOT a closed interval. Can you think of a function that is differentiable on (a, b) but NOT continuous on [a, b]? Of course, that depends precisely upon the difference between (a, b) and [a, b].

3. Nov 16, 2008

### haya

yes I mean closed interval [a,b]

for example F(x)= 1/x on [0,2]

F(x) is discontinuos at x=0

F(x) is differentiable on (0,2)

and thank you so much

4. Nov 17, 2008

### Pere Callahan

This example does not satisfy (iii) since $?=F(0)\neq F(2)=1/2$

5. Nov 17, 2008

### HallsofIvy

Try F(x)= 1/x on (0, 2], F(0)= 1/2.

6. Nov 18, 2008

### haya

yes it is satisfyed the theorem

7. Nov 21, 2008

### haya

but we know if the function is not continuous then it is not diffrentiable

so the answer of question (3) is will be no example because he want the function be discontinuous on interval [a,b] and diffrentiable on (a,b) that's never ever will happen in mathematics

8. Nov 21, 2008

### Pere Callahan

The function in our example is continous everywhere except at zero. At zero however, it is not required to be differentiable, so no problem.

9. Nov 21, 2008

### haya

could you give other example :

the example will be in discontinuoty on [a,b]
but differentiable on (a,b)

ok

I will be greatfull for you if give me the example
because this qeustion from the homework and tomorrow is the due day

10. Nov 21, 2008

### Office_Shredder

Staff Emeritus
You're correct, to the point where if the function is differentiable on (a,b) it's continuous on (a,b). Hence the only discontinuities can be at the endpoint. Examples are usually just continuous functions on (a,b) with the endpoints f(a) and f(b) redefined. Such as (a=0,b=1)

f(x)=x on (0,1) f(0)=f(1)=1
f(x)=sin(x) on (0,1) f(0)=2 f(1)=-3
f(x)=ex on (0,1) f(0)=1 f(1)=0