What is the extreme f(x,y)=(xy)/(sqrt(x^2+y^2-25)) ?

The extreme f(x,y)=(xy)/(sqrt(x^2+y^2-25)) is Saddle(-5,-5),Saddle(-5,5),Saddle(5,-5),Saddle(5,5)

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extreme f(x,y)=(xy)/(sqrt(x^2+y^2-25))

extreme points

f (x, y) = \frac{x y}{x ^{2} + y ^{2} - 25}

Saddle (- 5, - 5), Saddle (- 5, 5), Saddle (5, - 5), Saddle (5, 5)

Solution steps

Find the critical points:

(- 5, - 5), (- 5, 5), (5, - 5), (5, 5)

\frac{\partial ^{2} f}{\partial x ^{2}} = - \frac{3 x y ( x ^{2} - 25 )}{( x ^{2} + y ^{2} - 25 ) ^{\frac{5}{2}}}

D (x, y) = \frac{- 75 x ^{4} y ^{2} - 625 x ^{4} - 75 x ^{2} y ^{4} + 625 x ^{2} y ^{2} + 31250 x ^{2} - 625 y ^{4} + 31250 y ^{2} - 390625}{( x ^{2} + y ^{2} - 25 ) ^{5}}

Check the sign of at each critical point

Check critical point

(- 5, - 5) :

Saddle

Check critical point

(- 5, 5) :

Saddle

Check critical point

(5, - 5) :

Saddle

Check critical point

(5, 5) :

Saddle

Saddle (- 5, - 5), Saddle (- 5, 5), Saddle (5, - 5), Saddle (5, 5)

f (x) = x^{4} + 4 x^{2} - 10

f (x) = - x^{2} + 6

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

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

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

What is the extreme f(x,y)=(xy)/(sqrt(x^2+y^2-25)) ?
The extreme f(x,y)=(xy)/(sqrt(x^2+y^2-25)) is Saddle(-5,-5),Saddle(-5,5),Saddle(5,-5),Saddle(5,5)