Use Lagrange multiplier to find absolute maximum and minimum Use Lagrange multiplier to find absolute maximum and minimum of $f(x,y) =x^2+xy+y^2, x^2+y^2 =8$. 
What i've done so far..  
$f_x = \lambda g_x \Rightarrow 2x+y =\lambda2x, \\f_y = \lambda g_y \Rightarrow x+2y = \lambda 2y,\\g(x,y) = x^2+y^2 -8 =0$ 
May I know how should i proceed from here?

You have to determine $x,y,\lambda$. One possible way: 
$$x = 2y \cdot (\lambda-1)\\
\stackrel{eq. 1}{\Rightarrow} 4y \cdot (\lambda-1) +y = 2 \lambda \cdot 2y \cdot (\lambda-1) \\
\Leftrightarrow y \cdot (8\lambda-3-4\lambda^2)=0$$
There are two cases:


*

*$y=0$: From the first equation follows $x=0$, thus $x^2+y^2=0 \not= 8$. Can't be true...

*$-4\lambda^2+8\lambda-3=0$: Determine $\lambda$. Afterwards solve $x^2+y^2=8$ by using $x = 2y \cdot (\lambda-1)$. 

