What is the extreme f(x)=x^2-y^2x-xy ?

The extreme f(x)=x^2-y^2x-xy is Saddle(0,-1),Saddle(0,0),Minimum(-1/8 ,-1/2)

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extreme f(x)=x^2-y^2x-xy

extreme points

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

Saddle (0, - 1), Saddle (0, 0), Minimum (- \frac{1}{8}, - \frac{1}{2})

Solution steps

Find the critical points:

(0, - 1), (0, 0), (- \frac{1}{8}, - \frac{1}{2})

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

D (x, y) = - 4 x - 4 y^{2} - 4 y - 1

Check the sign of at each critical point

Check critical point

(0, - 1) :

Saddle

Check critical point

(0, 0) :

Saddle

Check critical point

(- \frac{1}{8}, - \frac{1}{2}) :

Minimum

Saddle (0, - 1), Saddle (0, 0), Minimum (- \frac{1}{8}, - \frac{1}{2})

f (x) = - x^{3} + 6 x^{2} + 63 x

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

f (x) = 5 x^{2} + 10 x

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

f (x) = x^{3} + 5 x^{2} - 9 x + 1

What is the extreme f(x)=x^2-y^2x-xy ?
The extreme f(x)=x^2-y^2x-xy is Saddle(0,-1),Saddle(0,0),Minimum(-1/8 ,-1/2)