# Homework Help: Twice differentiable functions

1. Aug 12, 2008

### JamesF

Hi all. Having a little trouble on this week's problem set. Perhaps one of you might be able to provide some insight.

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

$$f:[a,b] \rightarrow \mathbb{R}$$ is continuous and twice differentiable on (a,b). If f(a)=f(b)=0 and f(c) > 0 for some $$c \in (a,b)$$ then $$\exists \gamma \in (a,b)$$ s.t. $$f \prime \prime (\gamma) < 0$$

2. Relevant equations
Rolle's Theorem, MVT, Intermediate Value Theorem

3. The attempt at a solution

I'm not really sure how to approach the problem. I'm assuming you would apply Rolle's theorem or the Mean Value Theorem, or perhaps the Intermediate Value Property to the problem in order to obtain the solution.

With those theorems we can infer that $$\exists \theta$$ st $$f \prime (\theta) = 0$$
f(c) > 0 so there must be points u,v st f'(u) > 0 and f'(v) < 0

but none of that really gives me any info on the second derivative, which is what I need. I'm sure I'm overlooking something simple as usual, but if anyone could point me in the right direction it would be greatly appreciated.

2. Aug 12, 2008

### Dick

Look at the MVT on the intervals [a,c] and [c,b]. So, yes, there is a u in (a,c) and a v in (c,b) such that f'(u)>0 and f'(v)<0. Now apply the MVT to f'(x) on the interval [u,v]. The second derivative is the derivative of the first.

3. Aug 12, 2008

### JamesF

thank you very much Dick. You've been extremely helpful as always. I was able to get the answer.
The text passage is a discussion from a homework help forum, where a student is seeking help with a problem related to twice differentiable functions, and a responder provides guidance on how to solve it.
no_question

the passage does not contain a direct mathematical question that requires a numerical or calculative answer. instead, it is a discussion about a proof or a theoretical problem in calculus. the student is seeking guidance on how to approach the problem, and the responder provides a method to solve it. there are no specific calculations or word problems that require a numerical answer.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# Homework Help: Twice differentiable functions

1. Aug 12, 2008

### JamesF

Hi all. Having a little trouble on this week's problem set. Perhaps one of you might be able to provide some insight.

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

$$f:[a,b] \rightarrow \mathbb{R}$$ is continuous and twice differentiable on (a,b). If f(a)=f(b)=0 and f(c) > 0 for some $$c \in (a,b)$$ then $$\exists \gamma \in (a,b)$$ s.t. $$f \prime \prime (\gamma) < 0$$

2. Relevant equations
Rolle's Theorem, MVT, Intermediate Value Theorem

3. The attempt at a solution

I'm not really sure how to approach the problem. I'm assuming you would apply Rolle's theorem or the Mean Value Theorem, or perhaps the Intermediate Value Property to the problem in order to obtain the solution.

With those theorems we can infer that $$\exists \theta$$ st $$f \prime (\theta) = 0$$
f(c) > 0 so there must be points u,v st f'(u) > 0 and f'(v) < 0

but none of that really gives me any info on the second derivative, which is what I need. I'm sure I'm overlooking something simple as usual, but if anyone could point me in the right direction it would be greatly appreciated.

2. Aug 12, 2008

### Dick

Look at the MVT on the intervals [a,c] and [c,b]. So, yes, there is a u in (a,c) and a v in (c,b) such that f'(u)>0 and f'(v)<0. Now apply the MVT to f'(x) on the interval [u,v]. The second derivative is the derivative of the first.

3. Aug 12, 2008

### JamesF

thank you very much Dick. You've been extremely helpful as always. I was able to get the answer.
user
The text passage is a discussion from a homework help forum, where a student is seeking help with a problem related to twice differentiable functions, and a responder provides guidance on how to solve it.

TYPE: NO_QUESTION

The passage does not contain a direct mathematical question that requires a numerical or calculative answer. Instead, it is a discussion about a proof or a theoretical problem in calculus. The student is seeking guidance on how to approach the problem, and the responder provides a method to solve it. There are no specific calculations or word problems that require a numerical answer.