What is the extreme f(x,y)=x^2y^2+y^2-4x^2+5y-3 ?

The extreme f(x,y)=x^2y^2+y^2-4x^2+5y-3 is Minimum(0,-5/2),Saddle(1/2 ,-2),Saddle(-1/2 ,-2)

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extreme f(x,y)=x^2y^2+y^2-4x^2+5y-3

extreme points

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

Minimum (0, - \frac{5}{2}), Saddle (\frac{1}{2}, - 2), Saddle (- \frac{1}{2}, - 2)

Solution steps

Find the critical points:

(0, - \frac{5}{2}), (\frac{1}{2}, - 2), (- \frac{1}{2}, - 2)

\frac{\partial ^{2} f}{\partial x ^{2}} = 2 x^{2} + 2

D (x, y) = - 12 x^{2} y^{2} - 16 x^{2} + 4 y^{2} - 16

Check the sign of at each critical point

Check critical point

(0, - \frac{5}{2}) :

Minimum

Check critical point

(\frac{1}{2}, - 2) :

Saddle

Check critical point

(- \frac{1}{2}, - 2) :

Saddle

Minimum (0, - \frac{5}{2}), Saddle (\frac{1}{2}, - 2), Saddle (- \frac{1}{2}, - 2)

f (x) = x^{3} - x^{2} - 12 + 2, 0 \leq x \leq 4

f (x) = \frac{x ^{2}}{x ^{2} + 108}

f (x) = 4 x + 5

f (x) = x - 3 ln (x)

f (x, y) = x + 2 y

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