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

The extreme-y^2+xy-x^3+y+3 is Saddle(1/3 ,-1/3),Maximum(3/4 , 1/2)

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

extreme points

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

Saddle (\frac{1}{3}, - \frac{1}{3}), Maximum (\frac{3}{4}, \frac{1}{2})

Solution steps

Find the critical points:

(\frac{1}{3}, - \frac{1}{3}), (\frac{3}{4}, \frac{1}{2})

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

D (y, x) = 12 x - 1

Check the sign of at each critical point

Check critical point

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

Saddle

Check critical point

(\frac{3}{4}, \frac{1}{2}) :

Maximum

Saddle (\frac{1}{3}, - \frac{1}{3}), Maximum (\frac{3}{4}, \frac{1}{2})

f (x, y) = - x^{4} + 8 x^{2} - y^{2} - 1

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

\frac{x + 5}{2 - x}

f (x) = x^{2} - 8

f (x) = \frac{- 2}{x ^{2} - 9}

What is the extreme-y^2+xy-x^3+y+3 ?
The extreme-y^2+xy-x^3+y+3 is Saddle(1/3 ,-1/3),Maximum(3/4 , 1/2)