Ahhhh so now we can finish
So we know at the horizontal tangents that the derivative is zero
Now using the equation from above we get
Now if we plug this back into the original equation we get
setting we get
so it happens when
So when x=1 we get
Now by the rational roots theorem we know the only possible rational roots are
If you check on y=1 works so (1,1) is a horizontal tangent
I will leave the other cases for you to check.
Note there may be other Real roots