anonymous
  • anonymous
Find the minimum and maximum values of the function subject to the given constraint:
Mathematics
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katieb
  • katieb
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anonymous
  • anonymous
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anonymous
  • anonymous
$$f(x,y)=2x+3y\\g(x,y)=x^2+y^2=4$$We know our extrema occur where the gradients are parallel (or anti-parallel):$$\nabla f(x,y)=(2,3)\\\nabla g(x,y)=(2x,2y)\\(2x,2y)=\lambda(2,3)\implies x=\lambda,y=\frac32\lambda$$Now let's solve using our constraint:$$\lambda^2+\frac94\lambda^2=4\\\frac{13}4\lambda^2=4\\\lambda^2=\frac{16}{13}\\\lambda=\pm\frac4{\sqrt{13}}$$... so we've found two extrema, \(\left(\dfrac4{\sqrt{13}},\dfrac6{\sqrt{13}}\right),\left(-\dfrac4{\sqrt{13}},-\dfrac6{\sqrt{13}}\right)\)
anonymous
  • anonymous
Where do you ge tthe Y coordinates after you get your lambda coordinate?

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anonymous
  • anonymous
@FutureMathProfessor \(y=\dfrac32\lambda=\pm\dfrac6{\sqrt{13}}\)

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