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

The extreme f(x,y)=ln(4-x^2-y^2)-sqrt(y) is Saddle(0,16.48180…)

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

extreme points

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

Saddle (0, 16.48180 \dots)

Solution steps

Find the critical points:

(0, 16.48180 \dots)

\frac{\partial ^{2} f}{\partial x ^{2}} = - \frac{2 ( 4 - x ^{2} + y ^{2} )}{( 4 - x ^{2} - y ^{2} ) ^{2}} + \frac{1}{4 y ^{\frac{3}{2}}}

D (x, y) = \frac{y ^{6} + x ^{2} y ^{4} - 12 y ^{4} - x ^{4} y ^{2} - 8 x ^{2} y ^{2} + 48 y ^{2} - 8 y ^{\frac{11}{2}} - 8 x ^{4} y ^{\frac{3}{2}} - 16 x ^{2} y ^{\frac{7}{2}} + 128 y ^{\frac{3}{2}} - x ^{6} + 4 x ^{4} + 16 x ^{2} - 64}{2 y ^{\frac{3}{2}} ( 4 - x ^{2} - y ^{2} ) ^{4}}

Check the sign of at each critical point

Check critical point

(0, 16.48180 \dots) :

Saddle

Saddle (0, 16.48180 \dots)

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

f (x, y) = x^{2} y - 5 x y^{3}

\frac{x + 4}{x - 3}

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

f (x, y) = 14 x^{2} - 2 x^{3} + 4 x y

What is the extreme f(x,y)=ln(4-x^2-y^2)-sqrt(y) ?
The extreme f(x,y)=ln(4-x^2-y^2)-sqrt(y) is Saddle(0,16.48180…)