Examine whether function f(x,y)=x^2*y^2+1-4(x^2+y^2) has any stationary points?

A function has stationary points if and only if the
following equations are fulfiled:

df/dx = 0 and df/dy =
0

We'll calculate the partial derivative, with respect to
x, assuming that y is a constant;

df/dx = 2y^2*x -
8x

df/dx = 2x(y^2 - 4)

We'll
put df/dx = 0 => 2x(y^2 - 4) = 0

We'll set each
factor as zero:

2x = 0

x =
0

y^2 - 4 = 0

y^2 = 4
=> y1 = 2 and y2 = -2

We'll calculate the partial
derivative df/dy, with respect to y, assuming that x is a
constant:

df/dy = 2x^2*y -
8y

df/dy = 0

2y(x^2 - 4) =
0

2y = 0 => y3 = 0

x^2
- 4 = 0

x^2 = 4

x2 = 2 and x3
= -2

We'll get 5 stationary points:(0,0) ;
(2,2) ; (2,-2) ; (-2,-2) ; (-2,2).