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   Maximum end point value (Posted on 2019-07-15)
  Suppose a function f: [0, 10] -> R is continuous and differentiable everywhere in its domain. If f(10) = 19, and |f'(x) - 5| ≤ 4 for all x in the domain, what is the maximum value of f(0)?
  
   No Solution Yet Submitted by Danish Ahmed Khan No Rating
  
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   solution Comment 1 of 1
  from
  |f'(x)-5|<=4
  -4<=f'(x)-5<=4
  1<=f'(x)<=9
  thus f'(x)>0 and is strictly increasing on [0,10]
  
  we are given f(10)=19, let f(0)=a, then from the mean value theorem we have that there exists a value x on [0,10] for which f'(x)=[f(10)-f(0)]/(10-0)=(19-a)/10 but from above we know this must satisfy
  1<=f'(x)<=9
  Thus
  1<=(19-a)/10<=9
  10<=19-a<=90
  -171<=a<=71
  thus the maximum possible value for f(0) is 9
  
  For an explicit example take f(x)=9+x
  f(0)=9, f(10)=19 and f'(x)=1 which satisfies 1<=f'(x)<=9
  
   Posted by Daniel on 2019-07-15 12:44:04
  
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explanation: The text extract contains a math question posted on a website called perplexus.info. The question is about finding the maximum value of a function at a specific point given certain conditions on the function and its derivative. The extract also includes a solution provided by a user named Daniel, which explains the reasoning and calculations to determine the maximum value of \( f(0) \).
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars
   perplexus dot info
  
   Maximum end point value (Posted on 2019-07-15)
  Suppose a function f: [0, 10] -> R is continuous and differentiable everywhere in its domain. If f(10) = 19, and |f'(x) - 5| ≤ 4 for all x in the domain, what is the maximum value of f(0)?
  
   No Solution Yet Submitted by Danish Ahmed Khan No Rating
  
  Comments: ( Back to comment list | You must be logged in to post comments.)
   solution Comment 1 of 1
  from
  |f'(x)-5|<=4
  -4<=f'(x)-5<=4
  1<=f'(x)<=9
  thus f'(x)>0 and is strictly increasing on [0,10]
  
  we are given f(10)=19, let f(0)=a, then from the mean value theorem we have that there exists a value x on [0,10] for which f'(x)=[f(10)-f(0)]/(10-0)=(19-a)/10 but from above we know this must satisfy
  1<=f'(x)<=9
  Thus
  1<=(19-a)/10<=9
  10<=19-a<=90
  -171<=a<=71
  thus the maximum possible value for f(0) is 9
  
  For an explicit example take f(x)=9+x
  f(0)=9, f(10)=19 and f'(x)=1 which satisfies 1<=f'(x)<=9
  
   Posted by Daniel on 2019-07-15 12:44:04
  
   Search: Search body:
  Forums (1)
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The text extract contains a math question posted on a website called perplexus.info. The question is about finding the maximum value of a function at a specific point given certain conditions on the function and its derivative. The extract also includes a solution provided by a user named Daniel, which explains the reasoning and calculations to determine the maximum value of \( f(0) \).
  
  Label: question-and-answer