What is the extreme f(x,y)=48xy-32x^3-24y^2 ?

The extreme f(x,y)=48xy-32x^3-24y^2 is Saddle(0,0),Maximum(1/2 , 1/2)

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extreme f(x,y)=48xy-32x^3-24y^2

extreme points

f (x, y) = 48 x y - 32 x^{3} - 24 y^{2}

Saddle (0, 0), Maximum (\frac{1}{2}, \frac{1}{2})

Solution steps

Find the critical points:

(0, 0), (\frac{1}{2}, \frac{1}{2})

\frac{\partial ^{2} f}{\partial x ^{2}} = - 48

D (x, y) = 9216 x - 2304

Check the sign of at each critical point

Check critical point

(0, 0) :

Saddle

Check critical point

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

Maximum

Saddle (0, 0), Maximum (\frac{1}{2}, \frac{1}{2})

f (x, y) = x^{2} + x y + y^{2} - 34 y + 385

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

f (x) = x - 2 x

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

f (x, y) = 2 x^{2} + y^{2} + 3 x y - 3 y - 5 x + 8

What is the extreme f(x,y)=48xy-32x^3-24y^2 ?
The extreme f(x,y)=48xy-32x^3-24y^2 is Saddle(0,0),Maximum(1/2 , 1/2)