What is the extreme f(x,y)=x^3+3xy^2-15x^2-15y^2+72x ?

The extreme f(x,y)=x^3+3xy^2-15x^2-15y^2+72x is Minimum(6,0),Maximum(4,0),Saddle(5,1),Saddle(5,-1)

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extreme f(x,y)=x^3+3xy^2-15x^2-15y^2+72x

extreme points

f (x, y) = x^{3} + 3 x y^{2} - 15 x^{2} - 15 y^{2} + 72 x

Minimum (6, 0), Maximum (4, 0), Saddle (5, 1), Saddle (5, - 1)

Solution steps

Find the critical points:

(6, 0), (4, 0), (5, 1), (5, - 1)

\frac{\partial ^{2} f}{\partial x ^{2}} = 6 x - 30

D (x, y) = (6 x - 30)^{2} - 36 y^{2}

Check the sign of at each critical point

Check critical point

(6, 0) :

Minimum

Check critical point

(4, 0) :

Maximum

Check critical point

(5, 1) :

Saddle

Check critical point

(5, - 1) :

Saddle

Minimum (6, 0), Maximum (4, 0), Saddle (5, 1), Saddle (5, - 1)

f (x, y) = 10 - 4 x - 5 y

f (x, y) = x^{2} - \frac{1}{2} y^{2} + 3 x

f (x, y) = x^{2} - 3 x y + 6 y^{2} + 5 x - 2 y + 8

f (x) = x^{3} - 3 x^{2} + 4 x + 5

f (x) = cos (x) - sin (x)

What is the extreme f(x,y)=x^3+3xy^2-15x^2-15y^2+72x ?
The extreme f(x,y)=x^3+3xy^2-15x^2-15y^2+72x is Minimum(6,0),Maximum(4,0),Saddle(5,1),Saddle(5,-1)