Popular >

extreme f(x,y)=x^2-x^2y^2+y^2

Solution

extreme points

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

Minimum (0, 0), Saddle (1, - 1), Saddle (- 1, - 1), Saddle (1, 1), Saddle (- 1, 1)

Solution steps

Find the critical points:

(0, 0), (1, - 1), (- 1, - 1), (1, 1), (- 1, 1)

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

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

Check the sign of at each critical point

Check critical point

(0, 0) :

Minimum

Check critical point

(1, - 1) :

Saddle

Check critical point

(- 1, - 1) :

Saddle

Check critical point

(1, 1) :

Saddle

Check critical point

(- 1, 1) :

Saddle

Minimum (0, 0), Saddle (1, - 1), Saddle (- 1, - 1), Saddle (1, 1), Saddle (- 1, 1)

f (x, y) = 8 x y - x^{3} - 4 y^{2}

f (x, y) = 2 x^{2} + 3 y^{2} - 4 x - 5

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

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

f (x) = x \cdot ln (x)