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.