What is the extreme f(x,y)=3x^3-2x^2y+5xy-y^3-1 ?

The extreme f(x,y)=3x^3-2x^2y+5xy-y^3-1 is Saddle(0,0),Minimum(0.51868…,-0.82771…)

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extreme f(x,y)=3x^3-2x^2y+5xy-y^3-1

extreme points

f (x, y) = 3 x^{3} - 2 x^{2} y + 5 x y - y^{3} - 1

Saddle (0, 0), Minimum (0.51868 \dots, - 0.82771 \dots)

Solution steps

Find the critical points:

(0, 0), (0.51868 \dots, - 0.82771 \dots)

\frac{\partial ^{2} f}{\partial x ^{2}} = - 6 y

D (x, y) = - 108 x y + 24 y^{2} - 16 x^{2} + 40 x - 25

Check the sign of at each critical point

Check critical point

(0, 0) :

Saddle

Check critical point

(0.51868 \dots, - 0.82771 \dots) :

Minimum

Saddle (0, 0), Minimum (0.51868 \dots, - 0.82771 \dots)

f (x) = (3 - x) e^{x - 3}

f (x) = 1 + x^{2} e^{5 - 3 x}

y = - 4 e^{- 4 t} + 5 e^{- 5 t}

x^{3} + y^{3} + 9 x^{2} - 3 y^{2} + 15 x - 9 y

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

What is the extreme f(x,y)=3x^3-2x^2y+5xy-y^3-1 ?
The extreme f(x,y)=3x^3-2x^2y+5xy-y^3-1 is Saddle(0,0),Minimum(0.51868…,-0.82771…)