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

The extreme f(x,y)=3x^2y-y^2+5xy^2 is Saddle(4/15 ,-8/25)

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

extreme points

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

Saddle (\frac{4}{15}, - \frac{8}{25})

Solution steps

Find the critical points:

(\frac{4}{15}, - \frac{8}{25})

\frac{\partial ^{2} f}{\partial x ^{2}} = 10 x - 2

D (x, y) = - 100 y^{2} - 60 x y - 12 y - 36 x^{2}

Check the sign of at each critical point

Check critical point

(\frac{4}{15}, - \frac{8}{25}) :

Saddle

Saddle (\frac{4}{15}, - \frac{8}{25})

P (x, y) = 12 x^{6} - 7 x^{3} y - 10 y^{2}

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

f (x, y) = x^{3} - 6 x y + 8 y^{3}

f (x, y) = x^{2} + x^{2} y + y^{2} + 1

f (x) = 2 x + cos (x)

What is the extreme f(x,y)=3x^2y-y^2+5xy^2 ?
The extreme f(x,y)=3x^2y-y^2+5xy^2 is Saddle(4/15 ,-8/25)