What is the extreme f(x,y)=4xy^2-2x^2-16y^2 ?

The extreme f(x,y)=4xy^2-2x^2-16y^2 is Maximum(0,0),Saddle(4,2),Saddle(4,-2)

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extreme f(x,y)=4xy^2-2x^2-16y^2

extreme points

f (x, y) = 4 x y^{2} - 2 x^{2} - 16 y^{2}

Maximum (0, 0), Saddle (4, 2), Saddle (4, - 2)

Solution steps

Find the critical points:

(0, 0), (4, 2), (4, - 2)

\frac{\partial ^{2} f}{\partial x ^{2}} = 8 x - 32

D (x, y) = - 4 (8 x - 32) - 64 y^{2}

Check the sign of at each critical point

Check critical point

(0, 0) :

Maximum

Check critical point

(4, 2) :

Saddle

Check critical point

(4, - 2) :

Saddle

Maximum (0, 0), Saddle (4, 2), Saddle (4, - 2)

f (x) = \frac{3 x + 1}{1 - 2 x}

f (x, y) = 16 x + 12 y - 2 x^{2} - 3 y^{2}

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

x^{3} - 12 x + 3

f (x, y) = x y + \frac{3}{x} + \frac{9}{y}

What is the extreme f(x,y)=4xy^2-2x^2-16y^2 ?
The extreme f(x,y)=4xy^2-2x^2-16y^2 is Maximum(0,0),Saddle(4,2),Saddle(4,-2)