Prove that for all positive values of a and b 4a/b + b/a greater than or equal to 4

We are tasked with proving that for all positive values of \( a \) and \( b \), the following inequality holds:

\[
\frac{4a}{b} + \frac{b}{a} \geq 4
\]

### Step 1: Apply the Arithmetic Mean-Geometric Mean (AM-GM) Inequality
The Arithmetic Mean-Geometric Mean Inequality states that for any two non-negative real numbers \( x \) and \( y \):

\[
\frac{x + y}{2} \geq \sqrt{xy}
\]

We will now apply this inequality to the two terms in the expression, \( \frac{4a}{b} \) and \( \frac{b}{a} \).

### Step 2: Rewriting the expression
Let us set:

\[
x = \frac{4a}{b} \quad \text{and} \quad y = \frac{b}{a}
\]

We can now apply the AM-GM inequality to these two terms:

\[
\frac{x + y}{2} \geq \sqrt{xy}
\]

Substituting \( x \) and \( y \) back into the inequality:

\[
\frac{\frac{4a}{b} + \frac{b}{a}}{2} \geq \sqrt{\frac{4a}{b} \cdot \frac{b}{a}}
\]

### Step 3: Simplifying the inequality
Now, simplify the product inside the square root:

\[
\sqrt{\frac{4a}{b} \cdot \frac{b}{a}} = \sqrt{4} = 2
\]

Thus, we have:

\[
\frac{\frac{4a}{b} + \frac{b}{a}}{2} \geq 2
\]

Multiplying both sides by 2 gives:

\[
\frac{4a}{b} + \frac{b}{a} \geq 4
\]

### Conclusion
We have proven that for all positive values of \( a \) and \( b \):

\[
\frac{4a}{b} + \frac{b}{a} \geq 4
\]

### Summary:
This inequality follows from the application of the AM-GM inequality, which ensures that the sum of the terms is always greater than or equal to 4 for any positive values of \( a \) and \( b \).

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Would you like further details or have any questions?

Here are 5 related questions to deepen understanding:
1. Can you apply the AM-GM inequality to more than two terms?
2. How does the AM-GM inequality compare with the Cauchy-Schwarz inequality?
3. What happens if \( a \) and \( b \) are equal in the given inequality?
4. Can this technique be extended to prove inequalities involving more complex expressions?
5. How does the result change if \( a \) and \( b \) are negative?

**Tip**: The Arithmetic Mean-Geometric Mean (AM-GM) inequality is a powerful tool for solving inequalities and is often useful when terms involve both addition and multiplication.

Learn how to prove the inequality 4a/b + b/a >= 4 for all positive values of a and b using the Arithmetic Mean-Geometric Mean (AM-GM) Inequality. This step-by-step guide explains the process in detail and is suitable for students in Grades 10-12.

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