What is the extreme f(x,y)=sqrt(-4x^2+8x-9y^2-18y-12) ?

The extreme f(x,y)=sqrt(-4x^2+8x-9y^2-18y-12) is Maximum(1,-1)

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extreme f(x,y)=sqrt(-4x^2+8x-9y^2-18y-12)

extreme points

f (x, y) = - 4 x^{2} + 8 x - 9 y^{2} - 18 y - 12

Maximum (1, - 1)

Solution steps

Find the critical points:

(1, - 1)

\frac{\partial ^{2} f}{\partial x ^{2}} = - \frac{9 ( - 4 x ^{2} + 8 x - 3 )}{( - 4 x ^{2} + 8 x - 9 y ^{2} - 18 y - 12 ) - 9 y ^{2} - 18 y + 8 x - 4 x ^{2} - 12}

D (x, y) = \frac{36}{( 4 x ^{2} - 8 x + 9 y ^{2} + 18 y + 12 ) ^{2}}

Check the sign of at each critical point

Check critical point

(1, - 1) :

Maximum

Maximum (1, - 1)

f (x) = \frac{- x ^{3}}{3} - 2 x^{2} + 5 x - 2

f (x) = \frac{x}{4} + y - \frac{7}{4}

f (x, y) = \frac{1}{12} 144 - 16 x^{2} - 9 y^{2}

f (x) = 80 x^{2} - \frac{52}{3} x^{3} + x^{4}

f (x) = 2 x^{2} - 8 x + 8

What is the extreme f(x,y)=sqrt(-4x^2+8x-9y^2-18y-12) ?
The extreme f(x,y)=sqrt(-4x^2+8x-9y^2-18y-12) is Maximum(1,-1)