What is the extreme f(x,y)=32y^2+x^2-x^2y ?

The extreme f(x,y)=32y^2+x^2-x^2y is Minimum(0,0),Saddle(8,1),Saddle(-8,1)

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extreme f(x,y)=32y^2+x^2-x^2y

extreme points

f (x, y) = 32 y^{2} + x^{2} - x^{2} y

Minimum (0, 0), Saddle (8, 1), Saddle (- 8, 1)

Solution steps

Find the critical points:

(0, 0), (8, 1), (- 8, 1)

\frac{\partial ^{2} f}{\partial x ^{2}} = 64

D (x, y) = 64 (2 - 2 y) - 4 x^{2}

Check the sign of at each critical point

Check critical point

(0, 0) :

Minimum

Check critical point

(8, 1) :

Saddle

Check critical point

(- 8, 1) :

Saddle

Minimum (0, 0), Saddle (8, 1), Saddle (- 8, 1)

f (x) = x^{4} - 8 x^{3} + 22 x^{2} - 24 x

f (x) = - \frac{5 x ^{3}}{3} + 10 x^{2} + 25 x

f (x) = 8 - 3 x - 6 x^{2}

f (x) = 4 cos^{2} (x), 0 \leq x \leq π

cos (t)

What is the extreme f(x,y)=32y^2+x^2-x^2y ?
The extreme f(x,y)=32y^2+x^2-x^2y is Minimum(0,0),Saddle(8,1),Saddle(-8,1)