What is the extreme f(x,y)= 1/4 x^4-1/2 y^2-1/2 x^2-y+10 ?

The extreme f(x,y)= 1/4 x^4-1/2 y^2-1/2 x^2-y+10 is Maximum(0,-1),Saddle(-1,-1),Saddle(1,-1)

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extreme f(x,y)= 1/4 x^4-1/2 y^2-1/2 x^2-y+10

extreme points

f (x, y) = \frac{1}{4} x^{4} - \frac{1}{2} y^{2} - \frac{1}{2} x^{2} - y + 10

Maximum (0, - 1), Saddle (- 1, - 1), Saddle (1, - 1)

Solution steps

Find the critical points:

(0, - 1), (- 1, - 1), (1, - 1)

\frac{\partial ^{2} f}{\partial x ^{2}} = - 1

D (x, y) = - (3 x^{2} - 1)

Check the sign of at each critical point

Check critical point

(0, - 1) :

Maximum

Check critical point

(- 1, - 1) :

Saddle

Check critical point

(1, - 1) :

Saddle

Maximum (0, - 1), Saddle (- 1, - 1), Saddle (1, - 1)

f (x) = x^{3} + y^{3} - 27 x - 48 y - 24

f (x) = 4 (5 x y + \frac{1024}{y} + \frac{1024}{x})

f (x) = - 100 x^{2} + 1800 x

f (x) = \frac{1}{3} x^{3} - 2 x^{2} + 3 x - 4 [- 2.5]

- 9 x^{2} - 18 x + 432

What is the extreme f(x,y)= 1/4 x^4-1/2 y^2-1/2 x^2-y+10 ?
The extreme f(x,y)= 1/4 x^4-1/2 y^2-1/2 x^2-y+10 is Maximum(0,-1),Saddle(-1,-1),Saddle(1,-1)