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

The extreme f(x,y)=x^4+y^4+3x^2y+2x is Minimum(-1.44495…,-1.16124…)

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extreme f(x,y)=x^4+y^4+3x^2y+2x

extreme points

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

Minimum (- 1.44495 \dots, - 1.16124 \dots)

Solution steps

Find the critical points:

(- 1.44495 \dots, - 1.16124 \dots)

\frac{\partial ^{2} f}{\partial x ^{2}} = 12 y^{2}

D (x, y) = 12 y^{2} (12 x^{2} + 6 y) - 36 x^{2}

Check the sign of at each critical point

Check critical point

(- 1.44495 \dots, - 1.16124 \dots) :

Minimum

Minimum (- 1.44495 \dots, - 1.16124 \dots)

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

16

f (x) = - 5 + y - 10 x

f (x, y) = - 6 x^{2} + 7 x y - 2 y^{2} + x + y

f (x) = 4 x^{3} - 12 x^{2} + 9 x - 1

What is the extreme f(x,y)=x^4+y^4+3x^2y+2x ?
The extreme f(x,y)=x^4+y^4+3x^2y+2x is Minimum(-1.44495…,-1.16124…)