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 December 24th, 2013, 03:00 AM #1 Newbie   Joined: Dec 2013 Posts: 1 Thanks: 0 Local Extrema Find local extrema of f(x,y)= ((x^3)/3)+((y^2)/2)+xy-2x+4y+1
December 24th, 2013, 05:03 AM   #2
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Re: Local Extrema

Hello, crnogorac!

Quote:
 $\text{Find local extrema of: }\:f(x,y)\:=\:\frac{x^3}{3} \,+\, \frac{y^2}{2}\,+\,xy\,-\,2x\,+\,4y\,+\,1$

$\frac{\partial f}{\partial x} \:=\:x^2\,+\,y\,-\,2 \;=\;0\;\;[1]$

$\frac{\partial f}{\partial y} \:=\:y\,+\,x\,+\,4 \;=\;0 \;\;[2]$

$\text{From [2]: }\: y \:=\:-x\,-\,4$

$\text{Substitute into [1]: }\:x^2\,+\,(-x\,-\,4)\,-\,2 \:=\:0$

[color=beige]. . . [/color]$x^2\,-\,x\,-\,6 \:=\:0 \;\;\;\Rightarrow\;\;\;(x\,+\,2)(x\,-\,3) \:=\:0$

$\text{Therefore: }\:\begin{Bmatrix}x=&-2 \\ \\ \\ y=&-2\end{Bmatrix} \;\;\;\begin{Bmatrix}x=&3 \\ \\ \\ y=&-7\end{Bmatrix}=$

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