(a)The stronger hypothesis that y is a positive real number and x > y allows us to make some additional conclusions about x - y and (x - y)2. First, since x > y, when you subtract y from x (x - y), the result is also a positive real number. So, under this hypothesis, we can conclude that: 1 - y > 0: The difference x - y is greater than zero. Now, let's consider (x - y)2: The original proof states that (x - y)2 ≥ 0, which is true for any real numbers x and y. However, in this case, we can make a more specific observation: 2. (x - y)2 > 0: Given that x > y, the square of the positive difference (x - y) is strictly greater than zero. This is because a square of a positive non-zero number is always positive. So, the stronger hypothesis that x > y allows us to conclude that x - y is positive (x - y > 0), and the square of this difference (x - y)2 is strictly greater than zero ((x - y)2 > 0). (b) To prove that if x > y > 0, then , we can modify the proof as follows: Proof: Given the hypothesis that x > y > 0, we want to prove that . First, consider the difference x - y : Since x > y and both are positive real numbers, subtracting y from x gives us x - y, which is a positive real number because x is greater than y. 1. x - y > 0 Now, add 2xy to both sides of the inequality: x - y + 2xy > 2xy Then, we can factor in the left side of the inequality: x - y + 2xy = x(1+2y) - y Notice that x(1+2y) is always greater than y because we are given that x > y and 1 + 2y is greater than 1 since y is positive. So, we can write: x(1+2y) - y > y - y = 0 Thus: x(1+2y) - y > 0 Now, divide both sides of the inequality by 2: Now, we can rewrite this in terms of Since y > 0, we know that Now, we have proven that conclude: This completes the proof. and : . Therefore: , and since we are given that , we can (c) In the given proof, the step that relies on the fact that x and y are positive (although not explicitly stated) is when we take the square root of both sides of the inequality. This step is crucial because it assumes that the square root of x2 + 2xy + y2 is equal to x + y instead of -x - y because the square root of a positive number is always positive. identify the steps: Original Step: x2 + 2xy + y2 > 4xy Then, we rewrite it as: (x + y)2 > 4xy This step implicitly assumes that x and y are positive because taking the square root of x2 + 2xy + y2 yields x + y instead of -x - y. So, this step relies on the positivity of x and y, which was not explicitly mentioned but is a key assumption in the proof.