Problem 36 - Calculus with Concepts in Calculus

Question

Find the intervals on which the given Ir function is increasing and those on which it is decreasing. $$ f(x)=x|x| $$

Step-by-Step Solution

Answer

The function is increasing on \((0, \infty)\) and decreasing on \((-\infty, 0)\).

Step 1: Understand the Function

The function given is \( f(x) = x|x| \). Note that \(|x|\) denotes the absolute value of \(x\). Thus, \(f(x)\) can be rewritten as follows depending on the sign of \(x\): \(f(x) = x^2\) when \(x \geq 0\) and \(f(x) = -x^2\) when \(x < 0\).

Step 2: Differentiate the Piecewise Function

To determine where the function is increasing or decreasing, we need its derivative. The derivative of \( f(x) = x^2 \) for \( x \geq 0 \) is \( f'(x) = 2x \). For \( f(x) = -x^2 \) when \( x < 0 \), the derivative is \( f'(x) = -2x \).

Step 3: Analyze the Derivative for Intervals

For \( x \geq 0 \), the derivative \( f'(x) = 2x \), so the function is increasing when \( 2x > 0 \), i.e., \( x > 0 \). For \( x < 0 \), the derivative \( f'(x) = -2x \), so the function is increasing when \( -2x > 0 \) or \( x < 0 \). However, here it's decreasing because the inequality is not satisfied.

Step 4: Summarize Intervals of Increase/Decrease

Thus, \( f(x) \) is increasing on the interval \((0, \infty)\) and decreasing on the interval \((-\infty, 0)\). The function changes direction at \(x = 0\), which acts as a boundary point in the analysis but isn't included explicitly in either increasing or decreasing intervals.

Key Concepts

Piecewise Functions

Piecewise functions are a type of function constructed by piecing together different expressions. These expressions are defined over different intervals of the domain. When working with piecewise functions, it's crucial to understand that each piece applies to specific values of the variable. For example:

• For a function like \(f(x) = x|x|\), it behaves as \(x^2\) when \(x \geq 0\).
• When \(x < 0\), it takes the form \(-x^2\).

This is why these functions are called 'piecewise' — the pieces only govern specific parts of the input range. Understanding how and where each piece applies helps in further analysis like differentiation or evaluating limits.

Derivatives

Derivatives play a vital role in understanding the behavior of functions. They provide information about the rate at which a function's output value changes relative to changes in the input.
To find the derivative of a piecewise function like \(f(x) = x|x|\), you differentiate each distinct piece separately:

• For \(x \geq 0\), where \(f(x) = x^2\), the derivative is \(f'(x) = 2x\).
• For \(x < 0\), where \(f(x) = -x^2\), the derivative becomes \(f'(x) = -2x\).

By considering the derivative separately for each interval, we can understand how the function behaves across its entire domain. Differentiating correctly within each interval is crucial for determining where a function is increasing or decreasing.

Intervals of Increase and Decrease

To determine where a function increases or decreases, we analyze the sign of its derivative. When identifying intervals of increase and decrease:

• For \(x \geq 0\), the derivative \(2x > 0\) when \(x > 0\), implying the function is increasing.
• For \(x < 0\), the derivative \(-2x < 0\), means the function is decreasing.

In this exercise, \(f(x) = x|x|\) is increasing on the interval \((0, \infty)\) and decreasing on the interval \((-\infty, 0)\). Understanding these intervals helps plot and sketch the graph correctly and understand the function's trend over its domain.

Absolute Value Functions

Absolute value functions are essential in calculus as they transform any input into its non-negative value. In the context of \(f(x) = x |x|\), the absolute value helps in defining the function across different intervals:

• The absolute value \(|x|\) creates the piecewise nature of the function, differentiating behavior as \(x^2\) for non-negative \(x\) and \(-x^2\) for negative \(x\).
• This transformation allows the function to smoothly transition at \(x = 0\), without a jump or discontinuity, which is typical in functions involving absolute values.

Understanding the concept of absolute values is foundational for analyzing functions that behave differently in positive and negative domains, and it aids in predicting the slope and intercept at transition points like \(x=0\).