What is the extreme 48xy-32x^3-24y^2 ?

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

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extreme 48xy-32x^3-24y^2

extreme points

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) = 18 x y - x^{3} - 9 y^{2}

f (x) = 2 x^{3} - 27 x^{2} + 84 x + 9

f (x, y) = - 14 + 5 x^{2} + x y + y^{2}

f (x) = \frac{x ^{2}}{( x - 2 ) ^{2}}

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

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