Theory and Examples


In Exercises 51 and 52, give reasons for your answers.


Let f(x) = |x³ − 9x|.


a. Does f'(0) exist?

Verified step by step guidance

1. To determine if f'(0) exists, we need to check if the function f(x) = |x³ − 9x| is differentiable at x = 0. Differentiability requires the function to be continuous and have a defined derivative at that point.
2. First, check the continuity of f(x) at x = 0. Since f(x) is an absolute value function, it is continuous everywhere, including at x = 0.
3. Next, consider the definition of the derivative: f'(x) = lim (h -> 0) [(f(x + h) - f(x)) / h]. We need to evaluate this limit at x = 0.
4. To evaluate the derivative, consider the piecewise nature of the absolute value function. For x³ - 9x, identify the points where the expression inside the absolute value changes sign, which are the roots of x³ - 9x = 0.
5. Calculate the left-hand and right-hand derivatives at x = 0 by considering the limits from both sides. If these one-sided derivatives are equal, then f'(0) exists; otherwise, it does not.

Key Concepts

Derivative at a Point

The derivative of a function at a point measures the rate at which the function's value changes as its input changes. For f'(0) to exist, the function must be differentiable at x = 0, meaning it must be continuous and have a defined slope at that point. If the function has a sharp corner or cusp at x = 0, the derivative does not exist.

Absolute Value Function

The absolute value function, denoted as |x|, outputs the non-negative value of x. When applied to a function like f(x) = |x³ − 9x|, it can create points where the function is not smooth, such as cusps or corners, which can affect differentiability. Understanding how the absolute value impacts the function's graph is crucial for analyzing its derivative.

Piecewise Functions

A piecewise function is defined by different expressions over different intervals. The function f(x) = |x³ − 9x| can be expressed as a piecewise function, where the expression inside the absolute value changes sign. Analyzing the behavior of each piece separately helps determine the function's continuity and differentiability at critical points like x = 0.