(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.