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Rolles Theorem/ Mean Value Theorem + First Derivative Test

1. 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.

2. 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).


3. 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

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Homework Helper
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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).
 
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)=ex
 
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

Science Advisor
Homework Helper
4,284
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Are you saying that if a < b then f(a) cannot be equal to f(b) ?
 

HallsofIvy

Science Advisor
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1. 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.

2. 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).


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

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

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