Which one of the following inequalities is always true for positive real numbers x,y?

For positive real numbers x and y, an inequality is 'always true' only if it holds for all values in the domain (0, ∑). Option 1 (xy > x + y) fails for small values like x=1, y=1 (1 > 2 is false). Option 2 (x + y < (x + y)^2) fails when the sum is less than or equal to 1, such as x=0.2, y=0.3 (0.5 < 0.25 is false). Option 3 (x + y < x^2 + y^2) fails for x=0.5, y=0.5 (1 < 0.5 is false). Option 4 states 1 + x + y < (1 + x + y)^2. Let t = 1 + x + y. Since x, y > 0, it follows that t > 1. For any real number t > 1, the property t < t^2 always holds. This is because t^2 - t = t(t - 1), and if t > 1, both factors are positive, making the product positive.