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

The extreme x^2y-xy+3y^2 is Saddle(1,0),Saddle(0,0),Minimum(1/2 , 1/24)

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

extreme points

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

Saddle (1, 0), Saddle (0, 0), Minimum (\frac{1}{2}, \frac{1}{24})

Solution steps

Find the critical points:

(1, 0), (0, 0), (\frac{1}{2}, \frac{1}{24})

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

D (x, y) = 12 y - 4 x^{2} + 4 x - 1

Check the sign of at each critical point

Check critical point

(1, 0) :

Saddle

Check critical point

(0, 0) :

Saddle

Check critical point

(\frac{1}{2}, \frac{1}{24}) :

Minimum

Saddle (1, 0), Saddle (0, 0), Minimum (\frac{1}{2}, \frac{1}{24})

\frac{6}{x ^{2} + 2}

f (x) = 2 - x^{3}

\frac{6 x ^{2} - x ^{4}}{9}

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

f (x) = 7 x - 28 x

What is the extreme x^2y-xy+3y^2 ?
The extreme x^2y-xy+3y^2 is Saddle(1,0),Saddle(0,0),Minimum(1/2 , 1/24)