Problem 4 - Calculus Early Transcendentals: Pearson New International Edition

Question

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 everywhere in its domain.

Step 1: Find the Derivative

To determine where the function is increasing or decreasing, we first need to find its derivative. The derivative of the function \( f(x) = x^3 - 1 \) is \( f'(x) = 3x^2 \).

Step 2: Find Critical Points

Critical points occur where the derivative is zero or undefined. Set \( f'(x) = 3x^2 = 0 \) and solve for \( x \). The solution is \( x = 0 \). There are no points where the derivative is undefined since it is a polynomial.

Step 3: Determine Sign of Derivative

Examine the sign of \( f'(x) = 3x^2 \) to determine where the function is increasing or decreasing. Since \( 3x^2 \) is always non-negative and equals zero only at \( x = 0 \), \( f'(x) > 0 \) for all \( x eq 0 \).

Step 4: Conclusion on Monotonicity

Since the derivative \( f'(x) = 3x^2 \) is positive for all \( x eq 0 \), the function \( f(x) = x^3 - 1 \) is increasing on the entire set of real numbers except at the point \( x = 0 \). Since it does not decrease and is constant only at this single point, it is effectively increasing everywhere.

Key Concepts

Derivative

To understand the monotonicity of a function, calculating the derivative is crucial. The derivative provides a way to measure how a function changes as its input changes. For the function given, \( f(x) = x^3 - 1 \), its derivative is \( f'(x) = 3x^2 \). This involves applying basic differentiation rules: the power rule, where the derivative of \( x^n \) is \( nx^{n-1} \). Here, \( x^3 \) becomes \( 3x^2 \), showcasing how the derivative tells us the rate of change of \( f(x) \). The derivative is a key tool in analyzing a function's behavior.

Critical Points

Critical points are found where the derivative of a function is zero or undefined. These points are potential locations where the function could change its increasing or decreasing nature. For a polynomial like \( 3x^2 \), it’s critical to understand that the derivative is never undefined. Solving \( 3x^2 = 0 \) yields \( x = 0 \), pinpointing our critical point. This means \( x = 0 \) is a location where something notable happens in our function's slope, affecting its overall shape and behavior.

Polynomial

Polynomials are algebraic expressions consisting of variables and coefficients, involving terms with non-negative integer exponents. The function \( f(x) = x^3 - 1 \) is a polynomial of degree 3, where 'degree' signifies the highest power of the variable. The simplicity and continuity of polynomials make them good candidates for analysis using derivatives. A key property of polynomials is that they are smooth and continuous, meaning no sharp turns or breaks, and their derivatives exist everywhere on the real line.

Monotonic Function

A monotonic function is one that is entirely non-increasing or non-decreasing. By using the derivative, we can determine the monotonic nature of a function. For \( f(x) = x^3 - 1 \), the derivative \( f'(x) = 3x^2 \) is always non-negative, which means the function is non-decreasing. Since \( f'(x) > 0 \) for all \( x eq 0 \), the function is increasing wherever this condition holds. Thus, \( f(x) \) is increasing everywhere on the real numbers except at \( x = 0 \), a single point where it turns into a constant before increasing again. This understanding allows us to use the function's derivative to conclude its behavior across its domain.