# Local extreme value & saddle point: multi variable calculus

I am asked to find all local extreme values & saddle points of

$$f(x,y) = 2x^2 + y^2 - xy - 7y + 8$$

$$f_x(x, y) = 4x-y, \qquad f_y(x,y) = 2y-x-7$$

$$f_x(x,y) = 0, \qquad y = 4x$$

$$f_y(x,y) = 0, \qquad 2y-x-7 = 0, \qquad x = 1$$

So I have a critical point at $(1,4)$. Then I use 2nd derivative test to check min/max

$$f_{xx}(x,y) = 4, \qquad f_{yy}(x,y) = 2, \qquad f_{xy}(x,y) = -1$$

$$H(x,y) = 4\times 2 + (-1)^2 = 9$$

$H(1,4) > 0$, $f_{xx} > 0$ so local min. Answer given is "Local min $-6$ at $(1,4)$". What does $-6$ refer to?

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$$f(1,4)=2+16-4-28+8=26-32=-6$$