What is the extreme f(x,y)=3y^3+5x^2y-24x^2-24y^2-2 ?

The extreme f(x,y)=3y^3+5x^2y-24x^2-24y^2-2 is

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extreme f(x,y)=3y^3+5x^2y-24x^2-24y^2-2

extreme points

f (x, y) = 3 y^{3} + 5 x^{2} y - 24 x^{2} - 24 y^{2} - 2

Minimum (0, \frac{16}{3}), Maximum (0, 0), Saddle (\frac{24}{5 5}, \frac{24}{5}), Saddle (- \frac{24}{5 5}, \frac{24}{5})

Solution steps

Find the critical points:

(0, \frac{16}{3}), (0, 0), (\frac{24}{5 5}, \frac{24}{5}), (- \frac{24}{5 5}, \frac{24}{5})

\frac{\partial ^{2} f}{\partial x ^{2}} = 18 y - 48

D (x, y) = (10 y - 48) (18 y - 48) - 100 x^{2}

Check the sign of at each critical point

Check critical point

(0, \frac{16}{3}) :

Minimum

Check critical point

(0, 0) :

Maximum

Check critical point

(\frac{24}{5 5}, \frac{24}{5}) :

Saddle

Check critical point

(- \frac{24}{5 5}, \frac{24}{5}) :

Saddle

Minimum (0, \frac{16}{3}), Maximum (0, 0), Saddle (\frac{24}{5 5}, \frac{24}{5}), Saddle (- \frac{24}{5 5}, \frac{24}{5})

f (x) = sin (2 x), 0 \leq x \leq 2 π

f (x) = 2 x^{3} - 15 x^{2} + 36 x

f (x, y) = 60 x + 30 y - 2 x^{2} - 3 y^{2}

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

What is the extreme f(x,y)=3y^3+5x^2y-24x^2-24y^2-2 ?
The extreme f(x,y)=3y^3+5x^2y-24x^2-24y^2-2 is