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In calculus, when we discuss increasing intervals of a function, we refer to where the function's values are getting larger as we move from left to right along the x-axis. To determine these intervals, we often use the first derivative of the function.
The derivative, in this context, helps us find how the function is changing at any given point. If the derivative is positive (\(f'(x) > 0\)), it means that the function is climbing upwards at that point, indicating an increasing interval.
For example, consider the derivative \(f'(x) = x - 3\). To find where this derivative is positive, identify where \(x - 3 > 0\). Solve the inequality to find \(x > 3\). Therefore, the function \(f\) is increasing for all \(x\) values greater than 3. This segment of the function, known as an increasing interval, is denoted by \((3, \infty)\).
To ensure accuracy in identifying increasing intervals, you can choose a test point from this interval and substitute it back into the derivative. For instance, testing with \(x = 4\), \(f'(4) = 1\), which confirms that \(f\) is indeed increasing in \(x > 3\).

Just as we have increasing intervals, we also have decreasing intervals, which tell us where the function is declining as we move along the x-axis. Again, the derivative comes into play.
If the derivative \(f'(x)\) is negative (\(f'(x) < 0\)), the function is falling or decreasing over that interval. For our example, with the derivative \(f'(x) = x - 3\), set \(x - 3 < 0\) to find the decreasing intervals.
Solving this inequality gives \(x < 3\), which tells us the function \(f\) is decreasing when \(x\) is less than 3. This is expressed as the interval \((-\infty, 3)\).
For confirmation, choose a test value from this interval such as \(x = 2\). Substituting it in gives \(f'(2) = -1\), confirming the function is decreasing as the derivative here is negative.

Derivative analysis is a powerful tool in understanding the behavior of functions. At the core of this analysis is the process of taking the derivative of a function, which tells us the rate of change or slope at any given point.
To identify the nature of the function's behavior, we analyze the derivative's sign—positive, negative, or zero.

• If \(f'(x) > 0\), the function is increasing at that point.
• If \(f'(x) < 0\), the function is decreasing at that point.
• If \(f'(x) = 0\), we may have a critical point, indicating a potential maximum, minimum, or a point of inflection.

To illustrate, consider our derivative, \(f'(x) = x - 3\), used to find critical points by setting it equal to zero. Solving \(x - 3 = 0\) gives \(x = 3\). This critical point serves as a boundary separating increasing and decreasing behavior.
Along with determining intervals of increase or decrease, derivative analysis helps in sketching the graph and understanding the function's overall behavior. By combining derivative analysis with critical points, you build a comprehensive picture of how a function behaves across its domain.