What is the extreme x^2-2xy+y^3-y ?

The extreme x^2-2xy+y^3-y is Minimum(1,1),Saddle(-1/3 ,-1/3)

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extreme x^2-2xy+y^3-y

extreme points

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

Minimum (1, 1), Saddle (- \frac{1}{3}, - \frac{1}{3})

Solution steps

Find the critical points:

(1, 1), (- \frac{1}{3}, - \frac{1}{3})

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

D (x, y) = 12 y - 4

Check the sign of at each critical point

Check critical point

(1, 1) :

Minimum

Check critical point

(- \frac{1}{3}, - \frac{1}{3}) :

Saddle

Minimum (1, 1), Saddle (- \frac{1}{3}, - \frac{1}{3})

f (x) = 4 x + \frac{4}{x}

f (x) = 7 x^{4} - 6 x^{2} - 6

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

f (x) = 6 - 8 x, - 8 \leq x \leq 6

2 x^{5} - 15 x^{4} + 30 x^{3} - 6

What is the extreme x^2-2xy+y^3-y ?
The extreme x^2-2xy+y^3-y is Minimum(1,1),Saddle(-1/3 ,-1/3)