What is the extreme f(x,y)=xy(1-5x-10y) ?

The extreme f(x,y)=xy(1-5x-10y) is Saddle(0, 1/10),Maximum(1/15 , 1/30)

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extreme f(x,y)=xy(1-5x-10y)

extreme points

f (x, y) = x y (1 - 5 x - 10 y)

Saddle (0, \frac{1}{10}), Maximum (\frac{1}{15}, \frac{1}{30})

Solution steps

Find the critical points:

(0, \frac{1}{10}), (\frac{1}{15}, \frac{1}{30})

\frac{\partial ^{2} f}{\partial x ^{2}} = - 20 x

D (x, y) = - 100 x^{2} - 200 x y + 20 x - 400 y^{2} + 40 y - 1

Check the sign of at each critical point

Check critical point

(0, \frac{1}{10}) :

Saddle

Check critical point

(\frac{1}{15}, \frac{1}{30}) :

Maximum

Saddle (0, \frac{1}{10}), Maximum (\frac{1}{15}, \frac{1}{30})

f (x) = x^{2} + y^{2} + 18 x - 6 y

f (x) = 100 x + 100 y - x^{2} - y^{2}

f (x) = x^{2} + y^{2} + 18 x - 4 y

f (x) = x^{3} - 9 x^{2}, - 1 \leq x \leq 7

f (x, y) = x^{2} + y^{2} + 9

What is the extreme f(x,y)=xy(1-5x-10y) ?
The extreme f(x,y)=xy(1-5x-10y) is Saddle(0, 1/10),Maximum(1/15 , 1/30)