Problem 99 - Calculus of a Single Variable

Question

Proof Use the definitions of increasing and decreasing functions to prove that \(f(x)=x^{3}\) is increasing on \((-\infty, \infty) .\)

Step-by-Step Solution

Answer

Since \(x_1 < x_2\) implies \(x_1^3 < x_2^3\) for any real numbers \(x_1\) and \(x_2\), the function \(f(x)=x^3\) is increasing over the entire real line.

Step 1: Define the function and the property to prove

We are given the function \(f(x)=x^3\) and want to prove it is monotonically increasing over \((-\infty, \infty)\). This means for any two numbers \(x_1\) and \(x_2\) in the real line, if \(x_1 < x_2\), we need to show that \(f(x_1) < f(x_2)\).

Step 2: Proof

Let's assume that \(x_1 < x_2\). We then need to prove that \(f(x_1) < f(x_2)\), or equivalently that \(x_1^3 < x_2^3\). It is a property of real numbers that the cube of a bigger number is also bigger, thus it is true that if \(x_1 < x_2\), then \(x_1^3 < x_2^3\). Thus, the assumption leads to the conclusion, which proves that the function \(f(x)=x^3\) is increasing on the entire real line.

Key Concepts

Monotonic Functions

When exploring the behavior of functions in calculus, an important concept is that of monotonic functions. A function is said to be monotonic if it is consistently increasing or decreasing over its entire domain. This means there are no intervals where the function's behavior flips from increasing to decreasing or vice versa.

An **increasing function** steadily grows as you move along the x-axis, from left to right. In technical terms, for any two points, if the first is less than the second, then the image of the first under the function is less than or equal to the image of the second. Mathematically, for a function \(f(x)\), it is increasing on a domain if for any \(x_1, x_2\) such that \(x_1 < x_2\), we have \(f(x_1) \leq f(x_2)\).

An **increasing function** is **strictly increasing** if it meets the stronger condition \(f(x_1) < f(x_2)\) for \(x_1 < x_2\). Conversely, a **decreasing function** satisfies \(f(x_1) \geq f(x_2)\) for \(x_1 < x_2\), with strictness implying \(f(x_1) > f(x_2)\).

• Monotonic functions are predictable, making them easier to analyze.
• They don't have local maxima or minima within their interval of monotonicity.
• Knowing a function is monotonic helps in understanding its graph and limits.

Cube Function

The cube function, represented as \(f(x) = x^3\), is a type of polynomial function where each term's degree is three. This distinct function belongs to the power function family and possesses intriguing characteristics.

The **cube function** is defined for all real numbers, extending its applicability throughout the entire real line. It is a classic example of a strictly increasing function. This means that for any two real numbers, if the first is smaller than the second, then the cube of the first is also smaller than the cube of the second – formally, if \(x_1 < x_2\), then \(x_1^3 < x_2^3\).

Here are some of the function's key features:

• **Symmetry**: The cube function is an odd function, meaning its graph is symmetric about the origin. Thus, \(f(-x) = -f(x)\).
• **Point of Inflection**: At \(x = 0\), the function has a point of inflection, where the curvature changes but remains continuous.
• **End Behavior**: As \(x\) moves towards positive or negative infinity, the values of \(x^3\) grow or shrink accordingly, reflecting the function's limitless growth.

Proof in Calculus

In calculus, proofs provide a logically rigorous means of confirming mathematical truths. One common type of proof involves demonstrating the properties of functions, such as whether a function is increasing or decreasing across its domain.

The proof for the cube function \(f(x) = x^3\) being increasing over its entire domain uses basic principles of real numbers. Here's how it unfolds:

**Step-by-Step Approach**:

• Identify the domain of interest, here \((-\infty, \infty)\), where we want to prove monotonicity.
• Based on the calculus definition of a strictly increasing function, select any two points \(x_1\) and \(x_2\) such that \(x_1 < x_2\).
• Verify that the inequality \(x_1^3 < x_2^3\) holds true using the property that in real numbers, if \(a < b\), then \(a^3 < b^3\).

This kind of proof underpins much of calculus, combining logical thinking with algebraic manipulation to establish properties of mathematical entities. Understanding such proofs enhances grasp of not just specific functions, like the cube function, but the systematic reasoning behind many calculus-based analyses.