Introduction

This article delves into the process of finding the minimum value of the expression 2x^(3y) / (x^3 / y) where x and y are positive real numbers. The solution method is illustrated in a step-by-step manner, suitable for those familiar with basic calculus. The primary focus is on the application of partial derivatives and their use in optimization problems.

Setting Up the Problem

Consider the expression 2x^(3y) / (x^3 / y). Let's denote the value of this expression by z, where z is a function of x and y.

z 2x * (24y / x) * (3 / y).

First Partial Derivatives

To find the critical points, we need to find the first partial derivatives of z with respect to x and y, and set them to zero.

dz/dx 2 - (24y / x^2) dz/dy (24 / x) - (3 / y^2)

Solving for Critical Points

Setting these equal to zero gives us two equations:

2 - (24y / x^2) 0 (24 / x) - (3 / y^2) 0

By solving these equations, we can find the values of x and y that minimize the expression.

Elimination and Substitution

Clear the fractions and solve the equations for y and y^2: y (x^2) / 12 y^2 (x) / 8

Substituting the first equation into the second:

(x^4) / (2^4 * 3^2) (x) / (2^3) → x^3 18 → x (18)^(1/3)

Substitute this value of x back into the equation for y:

y ((18)^(1/3))^2 / 12.

Minimum Value Calculation

The minimum value, zmin, is calculated as follows:

zmin 2(18)^(1/3) * (212 * (18)^(2/3) / (18)^(1/3) * 12) * (312) / (18)^(2/3)) 6(18)^(1/3)

The approximate values are x ≈ 2.62074139421, y ≈ 0.572357121277, and zmin ≈ 15.72444383653.

Conclusion

The mathematical journey to finding the minimum value of the expression involves a combination of calculus and algebra. The final result not only provides insight into the optimization of the given function but also serves as a practical example for students and professionals in the field of mathematics and engineering.