What is the extreme f(x,y)=2x^3+xy^2+5x^2+y^2+6 ?

The extreme f(x,y)=2x^3+xy^2+5x^2+y^2+6 is Minimum(0,0),Maximum(-5/3 ,0),Saddle(-1,2),Saddle(-1,-2)

Popular >

extreme f(x,y)=2x^3+xy^2+5x^2+y^2+6

extreme points

f (x, y) = 2 x^{3} + x y^{2} + 5 x^{2} + y^{2} + 6

Minimum (0, 0), Maximum (- \frac{5}{3}, 0), Saddle (- 1, 2), Saddle (- 1, - 2)

Solution steps

Find the critical points:

(0, 0), (- \frac{5}{3}, 0), (- 1, 2), (- 1, - 2)

\frac{\partial ^{2} f}{\partial x ^{2}} = 2 x + 2

D (x, y) = (12 x + 10) (2 x + 2) - 4 y^{2}

Check the sign of at each critical point

Check critical point

(0, 0) :

Minimum

Check critical point

(- \frac{5}{3}, 0) :

Maximum

Check critical point

(- 1, 2) :

Saddle

Check critical point

(- 1, - 2) :

Saddle

Minimum (0, 0), Maximum (- \frac{5}{3}, 0), Saddle (- 1, 2), Saddle (- 1, - 2)

f (x) = x (6 - 2 x)^{2}

f (x, y) = 22 x^{4} + 22 y^{4} - 4 x y + 18

f (x) = \frac{1}{4} x^{4} + \frac{1}{3} x^{3} - x^{2} + \frac{2}{3}

f (x, y) = - 4 \cdot \frac{x ^{2} + 6 x + 8}{y ^{2} + 6 y + 6}

f (x) = x + \frac{108}{x ^{2}}

What is the extreme f(x,y)=2x^3+xy^2+5x^2+y^2+6 ?
The extreme f(x,y)=2x^3+xy^2+5x^2+y^2+6 is Minimum(0,0),Maximum(-5/3 ,0),Saddle(-1,2),Saddle(-1,-2)