Proving Monotonicity and Continuity of a Function

The study of function properties, such as monotonicity and continuity, is fundamental in mathematical analysis. This article explores a specific function and demonstrates its monotonic behavior through its derivatives and limits. We will also discuss how these properties can be extended to the closed interval.

Introduction

A function, f(x), is a relation between a set of inputs (the domain) and a set of permissible outputs where each input is related to exactly one output. Understanding the behavior of a function, such as whether it is increasing or decreasing, and whether it is continuous, is crucial in many areas of mathematics and its applications.

Function Definition and Initial Calculation

Let's consider the function defined as:

f(x) 2x2 log10(x) / (1 - x2) for the natural logarithm and similarly for the decimal logarithm.

First, let's calculate the value of the function at specific points:

Step 1: At x 1, we get:

f(1) 2 * 1 * log(1) / (1 - 12) 2 * 0 * 0 0.

Step 2: Derivative of the function for the natural logarithm:

f'(x) 2ln(x) / x - 2x.

Step 3: Derivative of the function for the decimal logarithm:

f'(x) 2log(x) / (x * ln(10)) - 2x.

In both cases, the derivative can be simplified as:

f'(x) 2ln(x) / x - 2x.

Evaluating the Function at Specific Points

Step 4: At x e (where e is the base of the natural logarithm), we get:

f'(e) 2 / e - 2e 2 / e - 2e ≈ -1.43656.

Step 5: At x e2, we get:

f'(e2) 2 / e2 - 2e2 ≈ -8.778.

Determining Monotonicity

The sign of the function's derivative, f'(x), determines whether the function is increasing or decreasing. From the above calculations, it is evident that f'(x) is negative for x > 1, indicating that the function is decreasing on this interval.

More formally, from the derivative, we can see that:

f''(x) 2 / x - 2 / x2.

For x ≥ 1, f''(x) is negative, confirming that f(x) decreases from 0 as x increases from 1 to infinity.

Therefore, f(x) is strictly decreasing on the infinite interval [1, ∞).

Continuity and Extension to the Closed Interval

A function is continuous if its limit exists and equals the value of the function at that point. The function f(x) is initially defined over the domain (1, ∞). We can extend this function by continuity to [0, ∞) by defining the limit as x approaches 0.

First, we evaluate the limit of the function as x approaches 0 from the right:

lim x→0 f(x) 2x ln(x) / (1 - x2).

This limit can be transformed into an indeterminate form 0 / 0, allowing us to apply L'Hopital's rule:

lim x→0 2 ln(x) / (1 / x) 2 * lim x→0 ln(x) / (1 / x) 2 * lim x→0 -1 / (1 / x2) 2 * 0 0.

Thus, the limit as x approaches 0 from the right is 1.

Final Remarks

The function f(x) is strictly decreasing on [1, ∞) and can be extended by continuity to [0, ∞). This analysis provides a clear and complete understanding of the function's behavior, addressing both its monotonicity and continuity.