http://calc3.askbot.com/questions/Open source question and answer forum written in Python and DjangoenCopyright Askbot, 2010-2011.Wed, 16 Jul 2014 20:04:41 -0500http://calc3.askbot.com/question/106/lagrange-multipliers-and-their-system-of-equations/On number 9 from the review sheet the question asks to find the extremes of $f(x, y) = 2x + 4y$ subject to $x^2+y^2=20$. When I was setting up my system of equations I got $$2=2\lambda x$$ $$4=2\lambda y$$ $$x^2+y^2=20$$ Then when I set them to 0 I got $$2-2\lambda x=0$$ $$4-2\lambda y=0$$ and when you simplify you are not able to pull out a variable $$2(1-\lambda x)=0$$ $$2(2-\lambda y)=0$$ My question is what do you do if this happens because the examples in my notes just have it where you are able to pull out a variable? *Comment*: Good question!!Wed, 16 Jul 2014 18:41:12 -0500http://calc3.askbot.com/question/106/lagrange-multipliers-and-their-system-of-equations/http://calc3.askbot.com/question/106/lagrange-multipliers-and-their-system-of-equations/?answer=107#post-id-107Given the system of equations: $$ 2 - 2 \lambda x = 0 $$ $$ 4 - 2 \lambda x = 0 $$ $$ x^2 + y^2 = 20 $$ We can simply solve for $x$ and $y$ in terms of $\lambda$ from the first two equations and plug it in to the third equation. Solving for $x$: $$ 2 - 2 \lambda x = 0 $$ $$ 2 \lambda x = 2 $$ $$ x = \frac{1}{\lambda} $$ Solving for $y$: $$ 4 - 2 \lambda y = 0 $$ $$ 2 \lambda y = 4 $$ $$ y = \frac{2}{\lambda} $$ Plugging in to the third equation: $$ \left(\frac{1}{\lambda}\right)^2 + \left(\frac{2}{\lambda}\right)^2 = 20 $$ $$ \frac{1}{\lambda^2} + \frac{4}{\lambda^2} = 20 $$ $$ \frac{5}{\lambda^2} = 20 $$ $$ \lambda^2 = \frac{5}{20} = \frac{1}{4} $$ $$ \lambda = \pm \frac{1}{2} $$ This means that there are $2$ cases to examine. When $\lambda = -\frac{1}{2}$: $$ x = \frac{1}{-\frac{1}{2}} = -2 $$ $$ y = \frac{2}{-\frac{1}{2}} = -4 $$ When $\lambda = \frac{1}{2}$: $$ x = \frac{1}{\frac{1}{2}} = 2 $$ $$ y = \frac{2}{\frac{1}{2}} = 4 $$ The two points that solve this system of equations are $(-2, -4)$ and $(2, 4)$. *Comment*: Awesome!!Wed, 16 Jul 2014 20:04:41 -0500http://calc3.askbot.com/question/106/lagrange-multipliers-and-their-system-of-equations/?answer=107#post-id-107