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Find he local maximum and minimum value and saddle points of the function: $$f(x,y)=x^2-xy+y^2-9x+6y+10$$

The answer is a min of $(-4,1), f(-4,1)=73$

I got a min of $(12/5,-21/5)$ my $$ f_x=2x-y-9\\ f_y=-x+2y+6 $$ set $f_x = 0 = f_y$ and we get $x=(9+y)/2$ so $y=-21/5$ and $x=12/5$

I found out that this was a minimum by the 2nd derivative test. There are no max or saddle points.

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You're both wrong (or maybe you made a mistake in copying $f$): the minimum is at $(4,-1)$ where the value is $-11$.

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You are solving the system, and your solution does not satisfy the 2 equations. The second is easier to use, you get $x=2y+6$ so $4y+12-y-9=0$, or equivalently, $y=-1$ and then $x = 6-2=4$...

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