Rolles Theorem/ Mean Value Theorem + First Derivative Test

[carlodelmundo]

Homework Statement

Suppose that f(x) is a twice-differentiable function defined on the closed interval [a,b]. If f'(c) = 0 for a < c < b, which of the following must be true?

I. f(a) = f(b)

II. f has a relative extremum at x = c.

III. f has a point of inflection at x = c.

Homework Equations

Rolles Theorem states that there is a c such that f'(c) = 0 between [a,b] if f(x) is continuous on [a,b] and differentiable on (a,b).

The Attempt at a Solution

By Rolles Theorem, statement "I" must be correct. (That is, the endpoints are equal to each other). Is this correct?

I also put statement "II" as correct because since f'(c) = 0 for a < c < b... c must be a critical point or relative extremum. Question though: does this mean there are other points between a and b where f'(c) = 0 ?

For statement "III", I said this was incorrect. It's twice differentiable yes, but we don't know if f(x) has a point of inflection at x = c.

Thanks!

 

[CompuChip]

You said: "if f'(c) = 0 for a < c < b."
Did you mean for all c between a and b, or for some c between a and b?

Because in the first case, for example I is true (although I think it follows from the principal theorem of analysis) while in the second case it is not (you can easily think of a counter example for some specific a, b, c and function f).

 

[aniketp]

Rolles Theorem states that there is a c such that f'(c) = 0 between [a,b] if f(x) is continuous on [a,b] and differentiable on (a,b).

I think there is one more condition here that f(a) = f(b)
Otherwise there is the counter example of f(x)=e^x

 

[carlodelmundo]

Thank you both. aniketp, you're right. f(a) = f(b) is the last condition... I overlooked it... and since a < b, they can't be equal to each other.

 

[CompuChip]

Are you saying that if a < b then f(a) cannot be equal to f(b) ?

 

[HallsofIvy]

Homework Statement

Suppose that f(x) is a twice-differentiable function defined on the closed interval [a,b]. If f'(c) = 0 for a < c < b, which of the following must be true?

I. f(a) = f(b)

II. f has a relative extremum at x = c.

III. f has a point of inflection at x = c.

Homework Equations

Rolles Theorem states that there is a c such that f'(c) = 0 between [a,b] if f(x) is continuous on [a,b] and differentiable on (a,b).

The Attempt at a Solution

By Rolles Theorem, statement "I" must be correct. (That is, the endpoints are equal to each other). Is this correct?

I also put statement "II" as correct because since f'(c) = 0 for a < c < b... c must be a critical point or relative extremum. Question though: does this mean there are other points between a and b where f'(c) = 0 ?

For statement "III", I said this was incorrect. It's twice differentiable yes, but we don't know if f(x) has a point of inflection at x = c.

Thanks!

As for "I", you are confusing the theorem with its converse. Rolle's theorem says that if f(a)= f(b) (and other conditions) then there exist c such that f'(c)= 0. Saying that some f'(c)= 0 does NOT means that f(a) must equal f(b).

Consider f(x)= x^2, a= -1, b= 2, c= 0. f'(0)= 0 but f(-1)\ne f(2). Or g(x)= x^3, a= -1, b= 2. Again g'(0)= 0 but g(-1)\ne g(2).

f has a relative extremum at 0 but not an inflection point. g has an inflection point at 0 but not a relative extremum. I would say none of those are necessarily true.

 

[carlodelmundo]

Thank you all! And yes... I did confuse Rolle's Theorem. Thanks for the heads up