How to Prove x ≤ xy/2 ≤ y for Real Numbers x and y
Introduction
This article aims to prove a fundamental inequality in real numbers: x ≤ xy/2 ≤ y given that x y. This involves a detailed step-by-step approach to understanding and proving the given inequality, along with its applications in trigonometric forms.
Proof of the Inequality
Given that x y, we need to prove that x ≤ xy/2 ≤ y.
Step 1: Proving x ≤ xy/2
First, let's consider the expression xy/2.
xy/2 x/2 y/2
Since it is given that x y, it follows that x/2 y/2. Therefore, adding the same terms to both sides, we get:
x/2 x/2 y/2 y/2
Which simplifies to:
x y/2 x/2
Substituting xy/2 x/2 y/2, we get:
x xy/2
To prove the equality part, we need to show that x xy/2 when x y. This is because both sides would then be equal to x y. Hence, the equality holds when x y but otherwise, x xy/2 is true.
Step 2: Proving xy/2 ≤ y
Now, to prove that xy/2 ≤ y, we can start from the expression:
xy/2 x/2 y/2
Since we know that y x/2 (from the previous inequality), then adding y/2 to both sides, we get:
y/2 y/2 ≥ x/2 y/2
This simplifies to:
y ≥ xy/2
Therefore, we have shown that:
x ≤ xy/2 ≤ y
This completes the proof for the inequality.
Mathematical Notation and Proof Using Inequalities
Formally, we can write the proof as follows:
Given: x y
Then:
x ≤ xy/2 ≤ y
Mathematically, this can be expressed as:
x y Rightarrow; x ≤ xy/2 ≤ y
Similarly, we can prove the other part:
xy/2 x/2 y/2
x/2 y/2 Rightarrow; x/2 x/2 y/2 y/2 Rightarrow; x xy/2
And
x/2 y/2 Rightarrow; x/2 y/2 ≥ y/2 y/2 Rightarrow; xy/2 ≥ y
Hence, we conclude:
x ≤ xy/2 ≤ y
Trigonometric Application Example
Given that x cos^(-1)α and y sin^(-1)α, the inequality x ≤ xy/2 ≤ y can be further explored.
Using these conditions, we can substitute:
xy cos^(-1)α sin^(-1)α
From the given condition, we get:
1/√2 ≤ α ≤ 1
Then, we calculate:
xy/2 (cos^(-1)α sin^(-1)α) / 2
Similarly:
xy/2 (π/2) / 2 π/4
Substitution back into the inequality, we obtain:
cos^(-1)α ≤ π/4 ≤ sin^(-1)α
This result is valid under the given conditions and provides an interesting application of the original inequality in a trigonometric context.
Conclusion
In this article, we have proven the inequality x ≤ xy/2 ≤ y for real numbers x, y, and explored its application in a trigonometric context. The proof involves detailed step-by-step analysis and the use of basic inequality properties. Understanding such inequalities is crucial in various fields, including mathematics and applications in real-life scenarios.