Problem 4 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-10, use the Monotonicity Theorem to find where the given function is increasing and where it is decreasing. \(f(x)=x^{3}-1\)

Step-by-Step Solution

Answer

The function is increasing on \((- fty, fty)\).

Step 1: Identify the Function and Find the Derivative

The given function is \( f(x) = x^3 - 1 \). To use the Monotonicity Theorem, we need to find the derivative of the function. The derivative \( f'(x) \) is calculated as follows:\[ f'(x) = \frac{d}{dx}(x^3 - 1) = 3x^2. \]

Step 2: Determine Critical Points

The critical points occur where the derivative is zero or undefined. Since \( f'(x) = 3x^2 \) and the derivative is defined everywhere, we set it to zero:\[ 3x^2 = 0. \]Solving this equation gives:\[ x = 0. \]Thus, \( x = 0 \) is the critical point.

Step 3: Analyze Intervals Using the Derivative

We need to analyze the signs of \( f'(x) \) on intervals determined by the critical points to determine where the function is increasing or decreasing. The critical point splits the domain into two intervals: \((-\infty, 0)\) and \((0, \infty)\).- For \( x \in (-\infty, 0) \), choose a test point like \( x = -1 \): \[ f'(-1) = 3(-1)^2 = 3 > 0. \] Thus, \( f(x) \) is increasing on \((-\infty, 0)\).- For \( x \in (0, \infty) \), choose a test point like \( x = 1 \): \[ f'(1) = 3(1)^2 = 3 > 0. \] Thus, \( f(x) \) is also increasing on \((0, \infty)\).

Step 4: Conclusion

Based on the sign of the derivative in each interval, we conclude that \( f(x) = x^3 - 1 \) is increasing for all \( x \in (-\infty, 0) \cup (0, \infty) \). The function does not decrease on any interval.

Key Concepts

Critical Points

Critical points are essential for understanding the behavior of functions. They occur at points where the derivative of a function is zero or undefined. In simpler terms, these are the points at which the function may change its direction of increase or decrease. In our exercise, we identified the function as \(f(x) = x^3 - 1\), and calculated its derivative to be \(f'(x) = 3x^2\). To find the critical points, we solve the equation \(3x^2 = 0\), which results in \(x = 0\).
Critical points help us divide the domain of the function into intervals, facilitating a detailed examination of the function's monotonic behavior. When analyzing a function, always start by finding its critical points as they can inform you where the function is increasing or decreasing.

Derivative

The derivative of a function is a fundamental concept in calculus, hinting at how the function changes at any point. It provides the slope of a function at any particular point. For the function \(f(x) = x^3 - 1\), the derivative \(f'(x) = 3x^2\) tells us how fast and in what direction the function is moving along the x-axis.
In practice, the derivative is crucial for detecting critical points and analyzing whether the function is increasing or decreasing in intervals. It's like a tool that allows us to zoom in on the fine details of a graph's shape, identifying where twists and turns might occur. Whenever you encounter a new function and need to analyze its behavior, finding the derivative should be your first step.

Increasing and Decreasing Functions

Understanding whether a function is increasing or decreasing is vital for interpreting its graph and real-world implications. A function is said to be increasing on an interval if, as you move from left to right along the interval, the function values rise. Conversely, it's decreasing if the function values fall.
The Monotonicity Theorem is a key tool in determining this behavior. For the given function \(f(x) = x^3 - 1\), by checking the sign of its derivative \(f'(x) = 3x^2\), we concluded that the function is increasing on both intervals \((-\infty, 0)\) and \((0, \infty)\). Because the derivative is consistently positive in these intervals, we see an upward trend, indicating the function is increasing throughout these ranges.

Calculus

Calculus is a branch of mathematics focused on studying change, with the derivative as one of its core concepts. Through calculus, we get precise tools to explore how functions behave under varying conditions. This study includes finding slopes of curves, optimizing values, and understanding the areas under curves.
In this exercise, we applied calculus—specifically, the concept of the derivative—to use the Monotonicity Theorem. This allows us to determine where the function \(f(x) = x^3 - 1\) is increasing. Calculus provides a broad framework for solving various mathematical problems, making it indispensable in both theoretical and applied contexts.