Proving (a^4b^4c^4d^4 geq 4abcd) Using the Arithmetic Mean-Geometric Mean Inequality

Mathematics often involves proving inequalities that at first glance can seem surprisingly complex. However, with the right tools, such as the Arithmetic Mean-Geometric Mean (AM-GM) inequality, these problems can become much more manageable. In this article, we will explore the proof that for any real numbers (a, b, c,) and (d), the inequality (a^4b^4c^4d^4 geq 4abcd) holds. This proof will be conducted using the AM-GM inequality, a fundamental concept in mathematical analysis.

Step-by-Step Proof Using the AM-GM Inequality

The AM-GM inequality states that for any set of non-negative real numbers (x_1, x_2, ldots, x_n), their arithmetic mean is greater than or equal to their geometric mean:

[frac{x_1 x_2 ldots x_n}{n} geq sqrt[n]{x_1x_2ldots x_n}]

The equality holds if and only if all (x_i) are equal.

Step 1: Applying the AM-GM Inequality

Let us denote (a^4, b^4, c^4,) and (d^4) as our set of non-negative real numbers. Applying the AM-GM inequality, we have:

[frac{a^4 b^4 c^4 d^4}{4} geq sqrt[4]{a^4b^4c^4d^4}]

Step 2: Simplifying the Inequality

We now need to simplify the right-hand side of the inequality. Notice that the fourth root of the product of the terms is the same as the fourth root of the product of the fourth powers:

[sqrt[4]{a^4b^4c^4d^4} (a^4b^4c^4d^4)^{frac{1}{4}} (abcd)^4 cdot (abcd)^{-3} abcd]

Thus, our inequality becomes:

[frac{a^4 b^4 c^4 d^4}{4} geq abcd]

Step 3: Formulating the Final Inequality

Multiplying both sides of the inequality by 4, we obtain the desired result:

[a^4b^4c^4d^4 geq 4abcd]

Conclusion

This rigorous proof using the AM-GM inequality demonstrates that for any real numbers (a, b, c,) and (d), the inequality (a^4b^4c^4d^4 geq 4abcd) holds true. Moreover, equality occurs if and only if (a b c d), providing a clear and precise condition for achieving the equality case.

It is critical to remember that the AM-GM inequality is a powerful tool in mathematical analysis and is often used to prove various inequalities and optimisation problems.

Examples to Consider

Let us consider an illustrative example to further solidify our understanding. Suppose (a 10) and (b c d 1). Plugging these values into our inequality, we have:

[a^4b^4c^4d^4 10^4 cdot 1^4 cdot 1^4 cdot 1^4 10000]

[4abcd 4 cdot 10 cdot 1 cdot 1 cdot 1 40]

Clearly, (10000 > 40), which confirms that the inequality (a^4b^4c^4d^4 geq 4abcd) holds.

Thus, the proof is not trivial when specific values are chosen, highlighting the strength and versatility of the AM-GM inequality.