What is the extreme f(x,y)=ln(9-x^2-y^2) ?

The extreme f(x,y)=ln(9-x^2-y^2) is Maximum(0,0)

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extreme f(x,y)=ln(9-x^2-y^2)

extreme points

f (x, y) = ln (9 - x^{2} - y^{2})

Maximum (0, 0)

Solution steps

Find the critical points:

(0, 0)

\frac{\partial ^{2} f}{\partial x ^{2}} = - \frac{2 ( 9 - x ^{2} + y ^{2} )}{( 9 - x ^{2} - y ^{2} ) ^{2}}

D (x, y) = - \frac{4 ( x ^{2} + y ^{2} + 9 )}{( x ^{2} + y ^{2} - 9 ) ^{3}}

Check the sign of at each critical point

Check critical point

(0, 0) :

Maximum

Maximum (0, 0)

y = 4 - x^{2}

2 cos (x) + sin (2 x)

f (x) = x^{3} - 18 x^{2} + 81 x + 45

2 x^{3} - 6 x^{2} - 18 x

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

What is the extreme f(x,y)=ln(9-x^2-y^2) ?
The extreme f(x,y)=ln(9-x^2-y^2) is Maximum(0,0)