text: # |(x^2 y^3)/(x^4+y^4 )| ≤|x|+|y|
  
  1
  
  2015-04-29T02:41:01+05:30
  
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  ( x² - y² )² >= 0
  x⁴ + y⁴ - 2 x² y² >= 0
  x⁴ + y⁴  >=  2 x² y²
  
  (x⁴ + y⁴) |y|  >=  2 x² y² |y|  >= 2 x² y³      ---- (1)
  
  further, we also see that  (x⁴ + y⁴) ( |y| + |x| )  >=  2 x² y³ + 2 x³ y²
  
  Thus we see clearly that,  from  (1)
  
  | (x² y³ ) / (x⁴ + y⁴) |  <=  |x| + |y|
explanation: The provided text is a mathematical proof or derivation that demonstrates the inequality \( \left| \frac{x^2 y^3}{x^4 + y^4} \right| \leq |x| + |y| \). The text does not contain a question or an answer to a question; it is a standalone mathematical argument.
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: # |(x^2 y^3)/(x^4+y^4 )| ≤|x|+|y|
  
  1
  
  2015-04-29T02:41:01+05:30
  
  ### This Is a Certified Answer
  
  Certified answers contain reliable, trustworthy information vouched for by a hand-picked team of experts. Brainly has millions of high quality answers, all of them carefully moderated by our most trusted community members, but certified answers are the finest of the finest.
  ( x² - y² )² >= 0
  x⁴ + y⁴ - 2 x² y² >= 0
  x⁴ + y⁴  >=  2 x² y²
  
  (x⁴ + y⁴) |y|  >=  2 x² y² |y|  >= 2 x² y³      ---- (1)
  
  further, we also see that  (x⁴ + y⁴) ( |y| + |x| )  >=  2 x² y³ + 2 x³ y²
  
  Thus we see clearly that,  from  (1)
  
  | (x² y³ ) / (x⁴ + y⁴) |  <=  |x| + |y|
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The provided text is a mathematical proof or derivation that demonstrates the inequality \( \left| \frac{x^2 y^3}{x^4 + y^4} \right| \leq |x| + |y| \). The text does not contain a question or an answer to a question; it is a standalone mathematical argument.
  
  Label: no-question