text: # Find the critical points of the following multivariable function, and using the Second Derivative Test to find the maxima and minima of the equation below: a(x,y)=(8x-y)/(e^(x^2+y^2)
  
  Find the critical points of the following multivariable function, and using the Second Derivative Test to find the maxima and minima of the equation below: $a\left(x,y\right)=\frac{8x-y}{{e}^{{x}^{2}+{y}^{2}}}$
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  hajavaF
  
  Differentiating the function,
  $a=\frac{8x}{{e}^{{x}^{2}+{y}^{2}}-\frac{y}{{e}^{{x}^{2}+{y}^{2}}}}$
  ${a}_{x}=\frac{8\left[{e}^{{x}^{2}+{y}^{2}}\cdot 1-x\cdot {e}^{{x}^{2}+{y}^{2}}\cdot 2x\right]}{{\left({e}^{{x}^{2}+{y}^{2}}\right)}^{2}}$
  ${a}_{x}=\frac{8\left[1-2{x}^{2}\right]}{{e}^{{x}^{2}+{y}^{2}}}$
  ${a}_{y}=\frac{-\left[{e}^{{x}^{2}+{y}^{2}}\cdot 1-y\cdot {e}^{{x}^{2}+{y}^{2}}\cdot yx\right]}{{\left({e}^{{x}^{2}+{y}^{2}}\right)}^{2}}$
  ${a}_{y}=\frac{-\left[1-2{y}^{2}\right]}{{e}^{{x}^{2}+{y}^{2}}}$
  For critical points, ${a}_{x}=0,{a}_{y}=0$
  ${a}_{x}=\frac{8\left[1-2{x}^{2}\right]}{{\left({e}^{{x}^{2}+{y}^{2}}\right)}^{2}}=0$
  $1-2{x}^{2}=0$
  $2{x}^{2}=1$
  $x=±\frac{1}{\sqrt{2}}$
  ${a}_{y}=\frac{-\left[1-2{y}^{2}\right]}{{\left({e}^{{x}^{2}+{y}^{2}}\right)}^{2}}=0$
  $1-2\frac{y}{62}=0$
  $2{y}^{2}=1$
  $y=±\frac{1}{\sqrt{2}}$
  The critical points are $\left(±\frac{1}{\sqrt{2}},±\frac{1}{\sqrt{2}}\right)$
  ${a}_{x}=\frac{8\left(1-2{x}^{2}\right)}{{e}^{{x}^{2}+{y}^{2}}}$
  ${a}_{×}=\frac{8\left(-6x+4{x}^{3}\right)}{{e}^{{x}^{2}+{y}^{2}}}$
  ${a}_{y}=\frac{-\left(1-2{y}^{2}\right)}{{e}^{{x}^{2}+{y}^{}}}$
explanation: The provided text contains a math question and a partial solution to the problem. The question asks for the critical points of a multivariable function and to use the Second Derivative Test to find the maxima and minima of the function. The text also includes a detailed step-by-step solution to find the critical points, but it does not complete the Second Derivative Test to determine the maxima and minima.
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: # Find the critical points of the following multivariable function, and using the Second Derivative Test to find the maxima and minima of the equation below: a(x,y)=(8x-y)/(e^(x^2+y^2)
  
  Find the critical points of the following multivariable function, and using the Second Derivative Test to find the maxima and minima of the equation below: $a\left(x,y\right)=\frac{8x-y}{{e}^{{x}^{2}+{y}^{2}}}$
  You can still ask an expert for help
  
  • Questions are typically answered in as fast as 30 minutes
  
  Solve your problem for the price of one coffee
  
  • Math expert for every subject
  • Pay only if we can solve it
  
  hajavaF
  
  Differentiating the function,
  $a=\frac{8x}{{e}^{{x}^{2}+{y}^{2}}-\frac{y}{{e}^{{x}^{2}+{y}^{2}}}}$
  ${a}_{x}=\frac{8\left[{e}^{{x}^{2}+{y}^{2}}\cdot 1-x\cdot {e}^{{x}^{2}+{y}^{2}}\cdot 2x\right]}{{\left({e}^{{x}^{2}+{y}^{2}}\right)}^{2}}$
  ${a}_{x}=\frac{8\left[1-2{x}^{2}\right]}{{e}^{{x}^{2}+{y}^{2}}}$
  ${a}_{y}=\frac{-\left[{e}^{{x}^{2}+{y}^{2}}\cdot 1-y\cdot {e}^{{x}^{2}+{y}^{2}}\cdot yx\right]}{{\left({e}^{{x}^{2}+{y}^{2}}\right)}^{2}}$
  ${a}_{y}=\frac{-\left[1-2{y}^{2}\right]}{{e}^{{x}^{2}+{y}^{2}}}$
  For critical points, ${a}_{x}=0,{a}_{y}=0$
  ${a}_{x}=\frac{8\left[1-2{x}^{2}\right]}{{\left({e}^{{x}^{2}+{y}^{2}}\right)}^{2}}=0$
  $1-2{x}^{2}=0$
  $2{x}^{2}=1$
  $x=±\frac{1}{\sqrt{2}}$
  ${a}_{y}=\frac{-\left[1-2{y}^{2}\right]}{{\left({e}^{{x}^{2}+{y}^{2}}\right)}^{2}}=0$
  $1-2\frac{y}{62}=0$
  $2{y}^{2}=1$
  $y=±\frac{1}{\sqrt{2}}$
  The critical points are $\left(±\frac{1}{\sqrt{2}},±\frac{1}{\sqrt{2}}\right)$
  ${a}_{x}=\frac{8\left(1-2{x}^{2}\right)}{{e}^{{x}^{2}+{y}^{2}}}$
  ${a}_{×}=\frac{8\left(-6x+4{x}^{3}\right)}{{e}^{{x}^{2}+{y}^{2}}}$
  ${a}_{y}=\frac{-\left(1-2{y}^{2}\right)}{{e}^{{x}^{2}+{y}^{}}}$
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The provided text contains a math question and a partial solution to the problem. The question asks for the critical points of a multivariable function and to use the Second Derivative Test to find the maxima and minima of the function. The text also includes a detailed step-by-step solution to find the critical points, but it does not complete the Second Derivative Test to determine the maxima and minima.
  
  Label: question-and-answer