text: # Search results
  
  1. ### Quadratic equation to find max and min
  
  Do you mean the proof? You can prove it through Lagrange multipliers, or their is a trigonometrical proof too. For the Lagrange multipliers proof, your optimization function is ##acos(\theta)+bsin(\theta)## and restrain function is ##cos^2(\theta)+sin^2(\theta)=1##.
  2. ### Quadratic equation to find max and min
  
  Also, if you ever need, ## - \sqrt{a^2+b^2} \leq acos(\theta)+bsin(\theta) \leq \sqrt{a^2+b^2}##
  3. ### Quadratic equation to find max and min
  
  It is very simple. Let us consider ##2x^{2} +y^{2}=4## Then ##\frac{x^{2}}{2}+\frac{y^{2}}{4}=1## Here we make the substitution ##\frac{x^2}{2}=cos^2(\theta)## and ##\frac{y^2}{4}=sin^2(\theta)## And that is how we get it.
  4. ### Quadratic equation to find max and min
  
  ##2x^2 + y^2 = 4## ##4x + y^2## I know you are done with the question, but I wanted to say that lagrange multipliers is not the best for this problem. Set ##x=\sqrt{2}cos(\theta), y=2sin(\theta)## So now we need to optimize ##4(\sqrt{2}cos(\theta)+sin^2(\theta) )##
explanation: The provided text extract consists of a series of posts or comments discussing the use of Lagrange multipliers and trigonometric substitutions to solve optimization problems involving quadratic equations. Each post builds on the previous one, providing different methods and insights. However, none of the posts explicitly contain a clear, standalone question or a question-and-answer pair.
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: # Search results
  
  1. ### Quadratic equation to find max and min
  
  Do you mean the proof? You can prove it through Lagrange multipliers, or their is a trigonometrical proof too. For the Lagrange multipliers proof, your optimization function is ##acos(\theta)+bsin(\theta)## and restrain function is ##cos^2(\theta)+sin^2(\theta)=1##.
  2. ### Quadratic equation to find max and min
  
  Also, if you ever need, ## - \sqrt{a^2+b^2} \leq acos(\theta)+bsin(\theta) \leq \sqrt{a^2+b^2}##
  3. ### Quadratic equation to find max and min
  
  It is very simple. Let us consider ##2x^{2} +y^{2}=4## Then ##\frac{x^{2}}{2}+\frac{y^{2}}{4}=1## Here we make the substitution ##\frac{x^2}{2}=cos^2(\theta)## and ##\frac{y^2}{4}=sin^2(\theta)## And that is how we get it.
  4. ### Quadratic equation to find max and min
  
  ##2x^2 + y^2 = 4## ##4x + y^2## I know you are done with the question, but I wanted to say that lagrange multipliers is not the best for this problem. Set ##x=\sqrt{2}cos(\theta), y=2sin(\theta)## So now we need to optimize ##4(\sqrt{2}cos(\theta)+sin^2(\theta) )##
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The provided text extract consists of a series of posts or comments discussing the use of Lagrange multipliers and trigonometric substitutions to solve optimization problems involving quadratic equations. Each post builds on the previous one, providing different methods and insights. However, none of the posts explicitly contain a clear, standalone question or a question-and-answer pair.
  
  Label: no-question