What is the extreme f(x,y)=ln(4-4x^2-y^2) ?

The extreme f(x,y)=ln(4-4x^2-y^2) is Maximum(0,0)

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extreme f(x,y)=ln(4-4x^2-y^2)

extreme points

f (x, y) = ln (4 - 4 x^{2} - y^{2})

Maximum (0, 0)

Solution steps

Find the critical points:

(0, 0)

\frac{\partial ^{2} f}{\partial x ^{2}} = - \frac{2 ( 4 - 4 x ^{2} + y ^{2} )}{( 4 - 4 x ^{2} - y ^{2} ) ^{2}}

D (x, y) = - \frac{16 ( 4 x ^{2} + y ^{2} + 4 )}{( 4 x ^{2} + y ^{2} - 4 ) ^{3}}

Check the sign of at each critical point

Check critical point

(0, 0) :

Maximum

Maximum (0, 0)

f (x) = \frac{8}{x ^{2} + 2}

f (x, y) = 2 x^{3} + 6 x y + 3 y^{2}

g (x, y) = x^{2} y^{3} + x^{4} y + 3 y^{2}

f (x, y) = - 2 x^{2} - y^{2} + 8 x + 10 y - 5 x y

f (x, y) = 4 x y^{2} - 2 x^{2} y - x

What is the extreme f(x,y)=ln(4-4x^2-y^2) ?
The extreme f(x,y)=ln(4-4x^2-y^2) is Maximum(0,0)