What is the extreme f(y,x)=(500)/(4+x^2+y^2) ?

The extreme f(y,x)=(500)/(4+x^2+y^2) is Maximum(0,0)

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extreme f(y,x)=(500)/(4+x^2+y^2)

extreme points

f (y, x) = \frac{500}{4 + x ^{2} + y ^{2}}

Maximum (0, 0)

Solution steps

Find the critical points:

(0, 0)

\frac{\partial ^{2} f}{\partial y ^{2}} = - \frac{1000 ( - 3 x ^{2} + y ^{2} + 4 )}{( 4 + x ^{2} + y ^{2} ) ^{3}}

D (y, x) = \frac{- 3000000 y ^{4} - 6000000 y ^{2} x ^{2} - 8000000 y ^{2} - 3000000 x ^{4} - 8000000 x ^{2} + 16000000}{( y ^{2} + x ^{2} + 4 ) ^{6}}

Check the sign of at each critical point

Check critical point

(0, 0) :

Maximum

Maximum (0, 0)

f (x) = \frac{x + 3}{x ^{2} - 4 x - 21}

2.5 x^{2} + 75 x + 25000

f (x) = (100 - 2 x) \cdot (75 - 2 x) \cdot x

f (x) = 5 x x - x^{2}

What is the extreme f(y,x)=(500)/(4+x^2+y^2) ?
The extreme f(y,x)=(500)/(4+x^2+y^2) is Maximum(0,0)