Mathematical Puzzles and Limits: Unraveling the Mystery of Squaring and Subtracting

Mathematics is a field where puzzles often lead to profound insights. One intriguing puzzle involves a step-by-step process that combines algebraic expression and limits. Let's delve into this fascinating problem and explore the mathematical concepts behind it.

Problem Statement

Imagine you have a number in your mind. Take this number, square it, then subtract the result from 108. Finally, multiply this difference by the original number. The question is, what is the maximum number you can obtain through these steps? Let's break down this problem and see where it leads us.

Algebraic Expression and Analysis

We start with the expression for our problem: x → x^2 → 108 - x^2 → (108 - x^2)x f(x). This expression represents the final value after all the operations are performed on the number x.

Exploring the Limit as x Approaches Negative Infinity

To understand the behavior of this expression, let's examine its limit as x approaches negative infinity. We can write this as:

lim_{x → -∞} f(x) lim_{x → -∞} (108 - x^2)x ∞

As x becomes more negative, the square term x^2 grows rapidly. However, when multiplied by a negative number, the result grows positively without bound. This means the function can achieve arbitrarily large positive values as x goes to negative infinity.

No Maximum Value for All Real Numbers

When we consider all real numbers without any restrictions, there is no upper limit to the value that f(x) can take. This is because the expression can grow without bound when x is sufficiently negative. Therefore, there is no maximum value for f(x) over all real numbers.

Considering Only Positive Numbers

However, what if we restrict ourselves to positive numbers? If the original number x must be non-negative, the behavior of the function changes. Let's rewrite the expression as:

y 108 - x^2 ? x

Graph this function to visualize its behavior. When x is positive, the term x^2 grows faster than the constant 108, which means that the expression 108 - x^3 will eventually become negative. As a result, the product with x will also become negative, indicating that the function has a maximum value.

Maximum Value for Positive Numbers

To find the maximum value, we can take the derivative of f(x) and set it to zero:

y' 108 - 3x^3 0

Solving for x gives us:

3x^3 108 → x^3 36 → x 36^{1/3}

Substitute this value back into the original expression to find the maximum value:

y 108 - (36^{1/3})^3 ? 36^{1/3} 108 - 36 ? 36^{1/3} 108 - 36^{4/3} 108 - 36 432

Thus, when x is a positive number, the maximum value of the expression is 432.

Conclusion

The behavior of the function reveals a critical difference between negative and positive numbers. While the function can theoretically grow without bound when x is negative, it has a finite maximum value of 432 when restricted to positive numbers.

This puzzle illustrates the importance of specifying conditions in mathematical problems. Without careful consideration, seemingly simple operations can lead to unexpected and limitless outcomes. By carefully defining the domain of the numbers involved, we can uncover specific and manageable solutions.