Optimizing Mathematical Expressions: Finding the Minimum Value of 3x y Given xy 10

In this article, we will explore the process of finding the minimum value of the mathematical expression (3x y) given the constraint (xy 10). We will employ techniques from calculus, specifically substitution and optimization, to solve this problem.

Introduction to the Problem

The problem at hand involves two positive real numbers, (x) and (y), such that (xy 10), and we are tasked with finding the minimum value of the expression (3x y).

Method of Substitution and Optimization

To solve this optimization problem, the first step is to express (y) in terms of (x) using the given constraint:

Step 1: Express (y) in terms of (x)

[ y frac{10}{x} ]

Step 2: Substitute (y) into the expression (3x y)

[ 3x y 3x frac{10}{x} ]

Step 3: Define the function to minimize

We define the function:

[ f(x) 3x frac{10}{x} ]

Finding the Derivative and Critical Points

To find the minimum value, we need to find the critical points of the function (f(x)) by taking its derivative and setting it to zero:

Step 4: Find the derivative of (f(x))

[ f'(x) 3 - frac{10}{x^2} ]

Step 5: Solve for (x) by setting the derivative equal to zero

[ 3 - frac{10}{x^2} 0 implies frac{10}{x^2} 3 implies x^2 frac{10}{3} implies x sqrt{frac{10}{3}} frac{sqrt{30}}{3} ]

Step 6: Find (y) using (y frac{10}{x})

[ y frac{10}{frac{sqrt{30}}{3}} frac{30}{sqrt{30}} sqrt{30} ]

Calculating the Minimum Value

Step 7: Substitute (x) and (y) back into the expression (3x y)

[ 3x y 3 left( frac{sqrt{30}}{3} right) sqrt{30} sqrt{30} sqrt{30} 2 sqrt{30} ]

The minimum value of (3x y) given (xy 10) is:

[ boxed{2 sqrt{30}} ]

Verifying the Minimum

To verify that this value is indeed a minimum, we check the second derivative:

Step 8: Find the second derivative of (f(x))

[ f''(x) frac{20}{x^3} ]

For (x > 0) and (x > 0), (f''(x) > 0), indicating that the function is concave up at (x sqrt{frac{10}{3}}). Therefore, this critical point is a local minimum.

Conclusion

We have rigorously determined that the minimum value of (3x y) given the constraint (xy 10) is (2 sqrt{30}). This exercise demonstrates the power of calculus in solving real-world optimization problems.