Maximizing (xyz) under the Constraint (x^2y^3z^4 26) Using the AM-GM Inequality

When dealing with optimization problems with constraints, one of the most common methods is to use the method of Lagrange multipliers or apply the Arithmetic Mean-Geometric Mean (AM-GM) inequality. In this article, we will explore how to use the AM-GM inequality to find the maximum value of (xyz) under the constraint (x^2y^3z^4 26).

Methodology

The AM-GM inequality states that for non-negative real numbers (a, b, c), the following holds true:

[frac{a b c}{3} geq sqrt[3]{abc}]

Step 1: Apply the AM-GM Inequality

We start by rewriting the given constraint (x^2y^3z^4 26). Let's introduce three new variables:

Let (a x^2) Let (b y^3) Let (c z^4)

Substituting these into the constraint, we get:

[a b c 26]

Step 2: Use AM-GM

Using the AM-GM inequality, we can write:

[frac{a b c}{3} geq sqrt[3]{abc}]

Substituting (a b c 26), we get:

[frac{26}{3} geq sqrt[3]{abc}]

Cubing both sides of the inequality:

[left(frac{26}{3}right)^3 geq abc]

Step 3: Relate (abc) to (xyz)

Differentiating (abc) in terms of (xyz), we have:

[abc x^2y^3z^4]

We can express (xyz) in terms of (a), (b), and (c):

[xyz sqrt{a} cdot sqrt[3]{b} cdot sqrt[4]{c}]

Step 4: Find Relationships

To find the maximum value of (xyz), we can use the substitutions for (x), (y), and (z) based on the constraint. Let:

(x sqrt{a}) (y b^{1/3}) (z c^{1/4})

Since (a x^2), (b y^3), and (c z^4), we can set:

(x^2 k) (y^3 k) (z^4 k)

For some (k). Therefore:

[k k k 26]

Which implies:

[3k 26]

And solving for (k):

[k frac{26}{3}]

Step 5: Calculate (xyz)

Now we can calculate:

[x sqrt{frac{26}{3}}, quad y left(frac{26}{3}right)^{1/3}, quad z left(frac{26}{3}right)^{1/4}]

Hence, the maximum value of (xyz) is:

[xyz sqrt{frac{26}{3}} cdot left(frac{26}{3}right)^{1/3} cdot left(frac{26}{3}right)^{1/4}]

Combining the exponents, we get:

[xyz left(frac{26}{3}right)^{frac{1}{2} frac{1}{3} frac{1}{4}}]

With a common denominator of 12, we find:

[frac{1}{2} frac{6}{12}, quad frac{1}{3} frac{4}{12}, quad frac{1}{4} frac{3}{12}]

Adding these together:

[frac{6 4 3}{12} frac{13}{12}]

Thus, the maximum value of (xyz) is:

[xyz left(frac{26}{3}right)^{frac{13}{12}}]

Conclusion

The maximum value of (xyz) under the constraint (x^2y^3z^4 26) is: