text: 1. ## local extrema
  
  Hi, I don't know how to find the local extrema, and I'm not sure if I differentiated correctly. for the first one is it -ln [(1-x)^2]-2? how do I find the critical point by letting it 0 =S would somebody be able to show me full solutions for part and b please?
  
  2. part b) $f(x) = cos(2x)+ sin(x)$
  
  $f'(x) = cos(x)- 2sin(2x)$
  
  for extreme value
  
  $0 = cos(x)-2sin(2x)$
  
  $0 = cos(x)-2\times 2sin(x)cos(x)$
  
  $0 = cos(x)-4sin(x)cos(x)$
  
  $0 = cos(x)(1-4sin(x))$
  
  $0 = 1-4sin(x)$
  
  $1 = 4sin(x)$
  
  $sin(x)=\frac{1}{4}$
  
  can you solve from here?
explanation: The document contains a math question and a partial solution to a related problem. The user is asking for help with finding local extrema and differentiating a function, and provides a detailed attempt at solving part of the problem.
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: 1. ## local extrema
  
  Hi, I don't know how to find the local extrema, and I'm not sure if I differentiated correctly. for the first one is it -ln [(1-x)^2]-2? how do I find the critical point by letting it 0 =S would somebody be able to show me full solutions for part and b please?
  
  2. part b) $f(x) = cos(2x)+ sin(x)$
  
  $f'(x) = cos(x)- 2sin(2x)$
  
  for extreme value
  
  $0 = cos(x)-2sin(2x)$
  
  $0 = cos(x)-2\times 2sin(x)cos(x)$
  
  $0 = cos(x)-4sin(x)cos(x)$
  
  $0 = cos(x)(1-4sin(x))$
  
  $0 = 1-4sin(x)$
  
  $1 = 4sin(x)$
  
  $sin(x)=\frac{1}{4}$
  
  can you solve from here?
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question and a partial solution to a related problem. The user is asking for help with finding local extrema and differentiating a function, and provides a detailed attempt at solving part of the problem.
  
  Label: question-and-answer