Question

Determine why Rolle's Theorem does not apply to the function f(x)= (2)/((x+1)^2) on the interval [-2,0].

Answer 1

Mind looking for and typing out the CONDITIONS that must be met for Rolle's Theorem to apply? You'll need to know these conditions and be able to check for them.

Answer 2

Ok so my teacher did not teach me this
so I looked it up
so f has to be continuous on the closed interval [-2,0]
f has to be differentiable on that interval
and f(a)=f(b)
is this right?
I just dont know how to prove these three things

Answer 3

Solomon? Unsure of what you are saying.

Answer 4

Look at the given function. Is it continuous on the interval [-2,0]? Differentiable? Explain. Does f(-2) = f(0)? Try evaluating this function for both x=-2 and x=0.

Answer 5

So
I have to
plug -2 and 0 into the original function?

Answer 6

At which point is \(\color{black}{\displaystyle f(x)=\frac{1}{(x+1)^2} }\) undefined?

Answer 7

At what value of \(\color{black}{\displaystyle x }\) do you get \(\color{black}{\displaystyle (x+1)^2=0 }\) ?

Answer 8

x=-1?

Answer 9

Yes, so that means that the function is undefined at \(\color{black}{\displaystyle x=-1 }\), correct?

Answer 10

Yes.

Answer 11

And one of your conditions to apply for Roll's Theorem is that the function has to be continuous over \(\color{black}{\displaystyle [-2,0]}\) ... Right?

Answer 12

Ohhh
yes.
So that means Rolle's Theorem does not apply because the function is not continues, it is undefined at x=-1.

Answer 13

continuous*

Answer 14

Yes, exactly.

Answer 15

Thank you very much.

Answer 16

NP:)
(I'm so used to thinking of Roll's Trm as just another case of Mean Value Trm, that I didn't even bother note that conditions for Roll's can be written differently. Sorry for that little inconvenience.)

Answer 17

Not to say that Roll's Trm is not another case of MVT, but as soon as I didn't see that "the slope between the endpoints" part, I assumed you left something out.

Answer 18

Yes. Conditions can be sticking points. Watch out for them and check any assumptions you make.

It's "Rolle's Theorem," by the way.

Answer 19

That's okay. @SolomonZelman

Answer 20

Sure:)

Answer 21

Gud luck!