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Find and classify the critical points of the function $$ f(x,y) = 5x^2 + 2xy + 5y^2. $$ Use the second derivative test to justify your answer.

For critical points I got $(0,0)$. Is that the only critical point?

Kenneth Hend
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2 Answers2

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$\nabla f=(10x+2y,2x+10y)=0\Rightarrow 5x+y=0 $ and $5y+x=0$

Therefore, correctly, the only critical point is (0,0).

Can you now classify it?

Dimitris
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  • It's a minimum. Is that right? – Kenneth Hend Apr 21 '13 at 00:34
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    Yes, taking the second derivatives : $f_{xx}=10,f_{yy}=10,f_{xy}=f_{yx}=2$. Now, $\Delta= f_{xx}f_{yy}-f^2_{xy}=96>0$, thus we have a local extremum. Since $f_{xx}>0 $, it is a local minimum , $f(0,0)=0$. – Dimitris Apr 21 '13 at 08:10
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What @Dimitris found is a critical point. Note as $f_{xx}=10,f_{yy}=10,f_{xy}=2$ so $\Delta_1=f_{xx}>0$ and $\Delta_2=f_{xx}f_{yy}-f^2_{xy}=100-4>0$ and therefore the origin is a local min for $f(x,y)$.

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Mikasa
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