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To understand where a function is increasing or decreasing, it's essential to examine its derivative, denoted as \( f'(x) \). The derivative provides information about the slope of the tangent line to the function at any point \(x\). If \( f'(x) > 0 \), the function is increasing on that interval. Conversely, if \( f'(x) < 0 \), the function is decreasing. Here, for \( f'(x) = x(x-1) \), we check the sign of \( f'(x) \) within specific intervals divided by the critical points.
When you solve \( x(x-1) = 0 \), you get the critical points \( x = 0 \) and \( x = 1 \). These points create intervals \((-\infty, 0)\), \((0, 1)\), and \((1, \infty)\). Analyzing each interval:

• On \((-\infty, 0)\), \( f'(x) > 0 \), indicating the function is increasing.
• On \((0, 1)\), \( f'(x) < 0 \), so the function is decreasing.
• On \((1, \infty)\), \( f'(x) > 0 \), showing an increasing function.

By applying this process, you can determine where the function's graph rises or falls.

Local maxima and minima refer to points where a function reaches its highest or lowest value, relative to nearby points. Using the critical points and changes in the increasing or decreasing behavior of the function, you can pinpoint these positions.
For our function, we found critical points at \( x = 0 \) and \( x = 1 \). The function's behavior changes around these points:

• At \( x = 0 \), the function shifts from increasing to decreasing, making \( x = 0 \) a local maximum. This indicates a peak where the function reaches a locally high value.
• At \( x = 1 \), the function switches from decreasing to increasing, making \( x = 1 \) a local minimum. Here, the function valley dips to a locally low value.

Identifying these shifts is crucial in understanding where a function adjusts from rising to falling or vice versa. This provides insight into the overall shape and behavior of the function's graph.

The First Derivative Test is a mathematical approach used to classify critical points based on the sign changes of the derivative \( f'(x) \). It helps determine whether a critical point is a local maximum, minimum, or neither. This test relies on the behavior of \( f'(x) \) before and after each critical point.
In our example:

• At \( x = 0 \), the derivative changes from positive to negative, indicating a local maximum. This transition shows the function peaks at this critical point.
• At \( x = 1 \), the derivative changes from negative to positive, showing a local minimum. This reversal represents the function reaching a bottom at this point.

By observing whether \( f'(x) \) changes from positive to negative (a local maximum) or from negative to positive (a local minimum), the First Derivative Test offers a methodical way of understanding the nature of critical points and their effect on the function's graph.