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extreme f(x,y)=xy(8+x)(y-9)

Solution

extreme points

f (x, y) = x y (8 + x) (y - 9)

Saddle (0, 0), Saddle (- 8, 0), Saddle (0, 9), Saddle (- 8, 9), Maximum (- 4, \frac{9}{2})

Solution steps

Find the critical points:

(0, 0), (- 8, 0), (0, 9), (- 8, 9), (- 4, \frac{9}{2})

\frac{\partial ^{2} f}{\partial x ^{2}} = 2 x (8 + x)

D (x, y) = - 12 x^{2} y^{2} + 108 x^{2} y - 324 x^{2} - 96 x y^{2} + 864 x y - 2592 x - 256 y^{2} + 2304 y - 5184

Check the sign of at each critical point

Check critical point

(0, 0) :

Saddle

Check critical point

(- 8, 0) :

Saddle

Check critical point

(0, 9) :

Saddle

Check critical point

(- 8, 9) :

Saddle

Check critical point

(- 4, \frac{9}{2}) :

Maximum

Saddle (0, 0), Saddle (- 8, 0), Saddle (0, 9), Saddle (- 8, 9), Maximum (- 4, \frac{9}{2})

\frac{3}{2} x^{2} + x y + \frac{1}{108} y^{4}

(3 x - 4)^{2}

\frac{x ^{3}}{x ^{2} + 4}

f (x) = 2 x^{3} + 3 x^{2} - 36 x - 20

x^{3} + 3 x^{2} y + y^{3} - y