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

The extreme f(x)=25(x^2-y^2)-2(x^2+y^2)^2 is Saddle(0,0),Maximum(-5/2 ,0),Maximum(5/2 ,0)

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extreme f(x)=25(x^2-y^2)-2(x^2+y^2)^2

extreme points

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

Saddle (0, 0), Maximum (- \frac{5}{2}, 0), Maximum (\frac{5}{2}, 0)

Solution steps

Find the critical points:

(0, 0), (- \frac{5}{2}, 0), (\frac{5}{2}, 0)

\frac{\partial ^{2} f}{\partial x ^{2}} = - 2 (25 + 4 x^{2} + 12 y^{2})

D (x, y) = 192 x^{4} + 384 x^{2} y^{2} + 800 x^{2} + 192 y^{4} - 800 y^{2} - 2500

Check the sign of at each critical point

Check critical point

(0, 0) :

Saddle

Check critical point

(- \frac{5}{2}, 0) :

Maximum

Check critical point

(\frac{5}{2}, 0) :

Maximum

Saddle (0, 0), Maximum (- \frac{5}{2}, 0), Maximum (\frac{5}{2}, 0)

f (x) = 2 x^{3} + 4 x^{2} + 2 x

f (x, y) = 80 x^{\frac{4}{10}} y^{\frac{2}{10}}

5 (x - 4)^{\frac{2}{3}}

f (x) = 5 x^{2} y + 3 x y^{8} + 23

f (x) = 5 x e^{- 4 x}

What is the extreme f(x)=25(x^2-y^2)-2(x^2+y^2)^2 ?
The extreme f(x)=25(x^2-y^2)-2(x^2+y^2)^2 is Saddle(0,0),Maximum(-5/2 ,0),Maximum(5/2 ,0)