What is the extreme f(x,y)=-4*(x^2+6x+8)/(y^2+6y+6) ?

The extreme f(x,y)=-4*(x^2+6x+8)/(y^2+6y+6) is Saddle(-3,-3)

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extreme f(x,y)=-4*(x^2+6x+8)/(y^2+6y+6)

extreme points

f (x, y) = - 4 \cdot \frac{x ^{2} + 6 x + 8}{y ^{2} + 6 y + 6}

Saddle (- 3, - 3)

Solution steps

Find the critical points:

(- 3, - 3)

\frac{\partial ^{2} f}{\partial x ^{2}} = - \frac{24 ( y ^{2} + 6 y + 10 ) ( x ^{2} + 6 x + 8 )}{( y ^{2} + 6 y + 6 ) ^{3}}

D (x, y) = \frac{- 64 x ^{2} y ^{2} - 384 x ^{2} y - 384 x ^{2} - 384 x y ^{2} - 2304 x y - 2304 x - 768 y ^{2} - 4608 y - 5376}{( y ^{2} + 6 + 6 y ) ^{4}}

Check the sign of at each critical point

Check critical point

(- 3, - 3) :

Saddle

Saddle (- 3, - 3)

What is the extreme f(x,y)=-4*(x^2+6x+8)/(y^2+6y+6) ?
The extreme f(x,y)=-4*(x^2+6x+8)/(y^2+6y+6) is Saddle(-3,-3)