Multiplying both given equations, (xy) + 1/(xy) = 29/10
Cross multiplying and rearranging, 10(xy)² - 29(xy) + 10 = 0
This is quadratic in (xy), whose graph is a parabola opening upwards. Hence it has only minimum defined value and maximum value is not defined.
Min defined value is the vertex of the parabola, which is given by (-b/2a, -D/4a).
-b/2a = 29/20
So evaluating with the function minimum value for xy = -11.25