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

The extreme f(x,y)=xy+x^2y+xy^2 is Saddle(0,-1),Maximum(-1/3 ,-1/3)

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

extreme points

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

Saddle (0, - 1), Maximum (- \frac{1}{3}, - \frac{1}{3})

Solution steps

Find the critical points:

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

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

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

Check the sign of at each critical point

Check critical point

(0, - 1) :

Saddle

Check critical point

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

Maximum

Saddle (0, - 1), Maximum (- \frac{1}{3}, - \frac{1}{3})

\frac{x ^{2} + 1}{x ^{3} - 1}

f (x) = - 17 \frac{x}{x ^{2} + y ^{2} + 1}

f (x) = x e^{- \frac{x}{4}}

f (x, y) = 350 \cdot x + 200 \cdot y

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