What is the extreme f(x,y)=2x^4-5x^2y^3-3xy^4 ?

The extreme f(x,y)=2x^4-5x^2y^3-3xy^4 is Saddle(-2048/3125 , 512/625)

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extreme f(x,y)=2x^4-5x^2y^3-3xy^4

extreme points

f (x, y) = 2 x^{4} - 5 x^{2} y^{3} - 3 x y^{4}

Saddle (- \frac{2048}{3125}, \frac{512}{625})

Solution steps

Find the critical points:

(- \frac{2048}{3125}, \frac{512}{625})

\frac{\partial ^{2} f}{\partial x ^{2}} = - 30 x^{2} y - 36 x y^{2}

D (x, y) = - 144 y^{6} - 360 x y^{5} - 600 x^{2} y^{4} - 864 x^{3} y^{2} - 720 x^{4} y

Check the sign of at each critical point

Check critical point

(- \frac{2048}{3125}, \frac{512}{625}) :

Saddle

Saddle (- \frac{2048}{3125}, \frac{512}{625})

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

f (x) = x^{3} - 9 x^{2} + 4

f (x) = \frac{1}{16 - x ^{2} + 4 y ^{2}}

f (x) = x^{3} (1 - x)^{2}

y = x^{\frac{2}{3}} (6 - x)^{\frac{1}{3}}

What is the extreme f(x,y)=2x^4-5x^2y^3-3xy^4 ?
The extreme f(x,y)=2x^4-5x^2y^3-3xy^4 is Saddle(-2048/3125 , 512/625)