Minimizing the Function f(x) x * (1/x) Without Calculus

In this article, we explore how to find the minimum value of the function f(x) x * (1/x) without using calculus. We will delve into graphical methods, algebraic techniques, and inequalities to achieve this goal. Our focus will be on positive values of x as the function is undefined at x 0.

Graphical Analysis

Let's start by examining the graph of the function f(x) x * (1/x). The function simplifies to f(x) 1 for all x ne; 0. However, when we consider values near 1, we can approximate the behavior around this point to identify any potential minimum.

By testing several values of x close to 1, we get:

f(0.6) approx; 2.27 f(0.7) approx; 2.13 f(0.8) 2.05 f(0.9) approx; 2.01 f(1.0) 2.00 f(1.1) approx; 2.01 f(1.2) approx; 2.03 f(1.3) approx; 2.07 f(1.4) approx; 2.11

From these values, it appears that the minimum value is at x 1.0, where f(1.0) 2. This suggests that there might be a local minimum at x 1.

Algebraic Techniques

To confirm this result, let's use algebraic techniques and inequalities. First, we note that for any positive value of x, the geometric mean of x and 1/x is always equal to 1:

[ text{Geometric Mean} sqrt{x cdot frac{1}{x}} 1 ]

By the Arithmetic Mean-Geometric Mean (AM-GM) inequality, the arithmetic mean (AM) of two numbers is always at least as large as their geometric mean (GM). Therefore, the AM of x and 1/x must be at least 2:

[ text{AM} frac{x frac{1}{x}}{2} geq 1 ]

For the AM to be at least 2:

[ frac{x frac{1}{x}}{2} geq 2 ]

Multiplying both sides by 2, we get:

[ x frac{1}{x} geq 4 ]

The only scenario where the AM equals the GM is when x 1/x. Solving for x in this equation, we get:

[ x^2 1 Rightarrow x 1 text{ (since } x > 0) ]

Thus, the minimum value of the function is:

[ f(1) 1 cdot frac{1}{1} 1 cdot 1 2 ]

This confirms that the minimum value of f(x) x * (1/x) is 2, and it is attained at x 1.

Harmonic Mean

In addition to the AM-GM inequality, let's consider the harmonic mean (HM). The HM of x and 1/x is at most their geometric mean (GM), which is 1:

[ text{Harmonic Mean} frac{2x cdot frac{1}{x}}{x frac{1}{x}} frac{2}{x frac{1}{x}} leq 1 ]

The HM is maximized when x 1/x, which again occurs when x 1. Hence, the minimum value of the function f(x) x * (1/x) is again 2 at x 1.

Conclusion

We have demonstrated that the minimum value of the function f(x) x * (1/x) is 2 and it is achieved at x 1. This conclusion is supported by both graphical analysis and algebraic techniques using the AM-GM inequality and the HM.

To summarize, we have:

Identified a local minimum using graphical methods. Confirmed the minimum value using the AM-GM inequality. Verified the result using the HM.

These methods provide a robust way to minimize the function without relying on calculus.