What is the extreme f(x,y)= 5/2 x^2+xy^2+5y^2 ?

The extreme f(x,y)= 5/2 x^2+xy^2+5y^2 is Minimum(0,0),Saddle(-5,5),Saddle(-5,-5)

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extreme f(x,y)= 5/2 x^2+xy^2+5y^2

extreme points

f (x, y) = \frac{5}{2} x^{2} + x y^{2} + 5 y^{2}

Minimum (0, 0), Saddle (- 5, 5), Saddle (- 5, - 5)

Solution steps

Find the critical points:

(0, 0), (- 5, 5), (- 5, - 5)

\frac{\partial ^{2} f}{\partial x ^{2}} = 2 x + 10

D (x, y) = 5 (2 x + 10) - 4 y^{2}

Check the sign of at each critical point

Check critical point

(0, 0) :

Minimum

Check critical point

(- 5, 5) :

Saddle

Check critical point

(- 5, - 5) :

Saddle

Minimum (0, 0), Saddle (- 5, 5), Saddle (- 5, - 5)

f (x) = 7 x^{3} - 7 x^{2} + 7

y = - \frac{4 x}{x ^{2} + 1}

f (x) = \frac{- 2}{3} x^{3} - 6 x^{2} - 10 x + 80

\frac{4 x + 5}{x - 7}

What is the extreme f(x,y)= 5/2 x^2+xy^2+5y^2 ?
The extreme f(x,y)= 5/2 x^2+xy^2+5y^2 is Minimum(0,0),Saddle(-5,5),Saddle(-5,-5)