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In differential calculus, understanding if a function is increasing or decreasing is crucial for analyzing its behavior. A function \( f(x) \) is called "increasing" on an interval if, as \( x \) moves to the right (towards greater numbers), the function's values also rise. Conversely, the function is "decreasing" if the function's values drop as \( x \) increases.
To determine this, we use the first derivative of the function, \( f'(x) \). Here are some key points to remember:

• If \( f'(x) > 0 \) on an interval, the function is increasing there.
• If \( f'(x) < 0 \) on an interval, the function is decreasing there.
• If \( f'(x) = 0 \), this might indicate a local maximum or minimum, or a point of inflection.

By analyzing the first derivative of a function we'll gather insights on where the original function climbs or dips. This is why knowing how to compute and interpret derivatives is incredibly useful. Be sure to split the function into parts when absolute values are involved, as these can change the function's behavior over different intervals.

Absolute value functions introduce a unique twist to basic function analysis. The absolute value, noted as \(|x|\), represents a number's distance from zero on the number line, stripping away any negative signs. This can significantly change how we approach finding increasing and decreasing intervals.
To simplify working with absolute value functions, break the function into "piecewise" functions where the expression inside the absolute value takes positive and negative values:

• For \( x - 1 \geq 0 \), the function is expressed without the absolute value as \( x - 1 \).
• For \( x - 1 < 0 \), rewrite \( |x - 1| \) as \( -(x - 1) \), reflecting over \( x \)-axis.

This split allows each part of the function to be analyzed separately. By taking these steps, the effects of the absolute value are managed, facilitating the calculation of derivatives and easy determination of increasing and decreasing intervals.

Derivatives are fascinating tools in calculus that help unpack the nuances of a function's growth or decline over specific intervals. The derivative, noted as \( f'(x) \), fundamentally represents the function's rate of change at any point. This calculation informs us of the slope of the function’s graph at particular values.
When assessing derivatives in relation to intervals:

• Compute the derivative of each segment of the function, particularly if split due to constants like absolute values.
• Test these derivatives for each of the intervals determined by the domains of the piecewise functions.
• Identify intervals where each derivative is positive or negative to understand where the original function rises or falls.

Analyzing derivatives gives clarity on the behavior of functions over the real number line. By combining the results for each piecewise interval, a comprehensive view of the function's overall behavior can be constructed.