text: # Thread: [SOLVED] curve sketching problem
  
  1. ## [SOLVED] curve sketching problem
  
  Consider the function $\displaystyle f(x) = e^{arctan(2x)}$
  
  1. Find the horizontal asymptotes of the function.
  2. State the interval(s) where the function is increasing.
  3. Find the inflection point.
  4. Find the intervals where the graph is concave up and concave down.
  
  It appears I've ran into a problem here. All I could do on my own so far was find f'(x):
  $\displaystyle f'(x) = e^{arctan(2x)}(\frac{2}{1+4x^2}) = \frac{2e^{arctan(2x)}}{1+4x^2}$
  
  Setting this equal to 0, $\displaystyle e^{arctan(2x)} = 0$ which has no solution? Doesn't look quite right to me.
  
  2. That's correct -- the function is monotonic over $\displaystyle \mathbb{R}$
  
  3. Ah, thank you! I should have noticed that just by looking at the function itself.
  
  I'm still not completely positive on whether or not I have done the rest of the problem correctly.
  This is what I have for the second derivative:
  
  $\displaystyle f''(x) = \frac{(1+4x^2)\frac{4e^{arctan(2x)}}{(1+4x^2)} - (2e^{arctan(2x)})(8x)}{(1+4x^2)^2}$
  
  Which then simplifies to be:
  
  $\displaystyle f''(x) = (\frac{2e^{arctan(2x)}}{(1+4x^2)^2})(2-8x)$
  
  I suppose the only way for it to equal zero would be x=1/4
  So the coordinates of the point of inflection are $\displaystyle (\frac{1}{4}, e^{arctan(1/2)})$
  
  And it would be concave up on $\displaystyle (-\infty,\frac{1}{4})$ and concave down on $\displaystyle (\frac{1}{4},\infty)$
  
  EDIT: Already submitted this answer and it turned out to be correct
explanation: The document is a discussion thread about a curve sketching problem involving the function \( f(x) = e^{\arctan(2x)} \). The user asks multiple questions related to the function, including finding horizontal asymptotes, intervals of increase, inflection points, and intervals of concavity. The user also provides their work and receives confirmation and additional insights from another user. The document contains multiple questions and answers.
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: # Thread: [SOLVED] curve sketching problem
  
  1. ## [SOLVED] curve sketching problem
  
  Consider the function $\displaystyle f(x) = e^{arctan(2x)}$
  
  1. Find the horizontal asymptotes of the function.
  2. State the interval(s) where the function is increasing.
  3. Find the inflection point.
  4. Find the intervals where the graph is concave up and concave down.
  
  It appears I've ran into a problem here. All I could do on my own so far was find f'(x):
  $\displaystyle f'(x) = e^{arctan(2x)}(\frac{2}{1+4x^2}) = \frac{2e^{arctan(2x)}}{1+4x^2}$
  
  Setting this equal to 0, $\displaystyle e^{arctan(2x)} = 0$ which has no solution? Doesn't look quite right to me.
  
  2. That's correct -- the function is monotonic over $\displaystyle \mathbb{R}$
  
  3. Ah, thank you! I should have noticed that just by looking at the function itself.
  
  I'm still not completely positive on whether or not I have done the rest of the problem correctly.
  This is what I have for the second derivative:
  
  $\displaystyle f''(x) = \frac{(1+4x^2)\frac{4e^{arctan(2x)}}{(1+4x^2)} - (2e^{arctan(2x)})(8x)}{(1+4x^2)^2}$
  
  Which then simplifies to be:
  
  $\displaystyle f''(x) = (\frac{2e^{arctan(2x)}}{(1+4x^2)^2})(2-8x)$
  
  I suppose the only way for it to equal zero would be x=1/4
  So the coordinates of the point of inflection are $\displaystyle (\frac{1}{4}, e^{arctan(1/2)})$
  
  And it would be concave up on $\displaystyle (-\infty,\frac{1}{4})$ and concave down on $\displaystyle (\frac{1}{4},\infty)$
  
  EDIT: Already submitted this answer and it turned out to be correct
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is a discussion thread about a curve sketching problem involving the function \( f(x) = e^{\arctan(2x)} \). The user asks multiple questions related to the function, including finding horizontal asymptotes, intervals of increase, inflection points, and intervals of concavity. The user also provides their work and receives confirmation and additional insights from another user. The document contains multiple questions and answers.
  
  Label: multiple-question-and-answer