What is the extreme f(x,y)=(6x-x^2)(4y-y^2) ?

The extreme f(x,y)=(6x-x^2)(4y-y^2) is Saddle(0,0),Saddle(6,0),Saddle(0,4),Saddle(6,4),Maximum(3,2)

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extreme f(x,y)=(6x-x^2)(4y-y^2)

extreme points

f (x, y) = (6 x - x^{2}) (4 y - y^{2})

Saddle (0, 0), Saddle (6, 0), Saddle (0, 4), Saddle (6, 4), Maximum (3, 2)

Solution steps

Find the critical points:

(0, 0), (6, 0), (0, 4), (6, 4), (3, 2)

\frac{\partial ^{2} f}{\partial x ^{2}} = - 2 (6 x - x^{2})

D (x, y) = - 12 x^{2} y^{2} + 48 x^{2} y - 64 x^{2} + 72 x y^{2} - 288 x y + 384 x - 144 y^{2} + 576 y - 576

Check the sign of at each critical point

Check critical point

(0, 0) :

Saddle

Check critical point

(6, 0) :

Saddle

Check critical point

(0, 4) :

Saddle

Check critical point

(6, 4) :

Saddle

Check critical point

(3, 2) :

Maximum

Saddle (0, 0), Saddle (6, 0), Saddle (0, 4), Saddle (6, 4), Maximum (3, 2)

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

ln (x) - 3 x + 2

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

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

- 3 x^{2} + 5 x y - 2 y^{2} + x + y

What is the extreme f(x,y)=(6x-x^2)(4y-y^2) ?
The extreme f(x,y)=(6x-x^2)(4y-y^2) is Saddle(0,0),Saddle(6,0),Saddle(0,4),Saddle(6,4),Maximum(3,2)