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extreme f(x)=(x^3-3x)(y+3)+y(y+6)

Solution

extreme points

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

Saddle (0, - 3), Saddle (- 3, - 3), Saddle (3, - 3), Minimum (- 1, - 4), Minimum (1, - 2)

Solution steps

Find the critical points:

(0, - 3), (- 3, - 3), (3, - 3), (- 1, - 4), (1, - 2)

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

D (x, y) = 12 x (y + 3) - 9 (x + 1)^{2} (x - 1)^{2}

Check the sign of at each critical point

Check critical point

(0, - 3) :

Saddle

Check critical point

(- 3, - 3) :

Saddle

Check critical point

(3, - 3) :

Saddle

Check critical point

(- 1, - 4) :

Minimum

Check critical point

(1, - 2) :

Minimum

Saddle (0, - 3), Saddle (- 3, - 3), Saddle (3, - 3), Minimum (- 1, - 4), Minimum (1, - 2)

4 h^{3} - 96 h^{2} + 527 h

f (x) = 25 x^{2} - 25 y^{2} + 200 x + 200 y + 200

x^{2} + y^{2} + 6 x - 10 y + 5

y = - 5 x^{3} + 4 x^{2} + 5 x - 1

f (x) = \frac{9 x ^{2} - 1}{x}