text: # 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.
explanation: The document is a forum post from a homework help thread discussing a problem involving twice differentiable functions. The post includes a problem statement, relevant equations, and an attempt at a solution. The original poster (JamesF) asks for help in solving the problem, and another user (Dick) provides guidance. JamesF then thanks Dick for the help and mentions that he was able to get the answer.
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distilabel_metadata.role: system
distilabel_metadata.content: # 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.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is a forum post from a homework help thread discussing a problem involving twice differentiable functions. The post includes a problem statement, relevant equations, and an attempt at a solution. The original poster (JamesF) asks for help in solving the problem, and another user (Dick) provides guidance. JamesF then thanks Dick for the help and mentions that he was able to get the answer.
  
  Label: question-and-answer