Increasing and Decreasing Intervals

1) What you are looking for

For a function, increasing intervals are where the output rises as the input moves left to right. Decreasing intervals are where the output falls as the input moves left to right.

2) Main method

1. Find the critical numbers if a derivative is available: solve where the derivative is 0 or undefined.
2. Split the number line at those points.
3. Test each interval with a value from that region.
4. Decide the sign of the derivative:
   • Positive derivative → function is increasing.
   • Negative derivative → function is decreasing.

3) Write the intervals clearly

Use interval notation and include only the parts where the function has the correct behavior. If the function changes behavior at a boundary, do not merge intervals with different signs.

4) Simplify the final answer

Combine adjacent intervals only when they describe the same behavior. Keep the answer exact and in clean interval notation.

5) Quick check

After finding the intervals, compare them with the sign changes of the derivative or with the graph’s overall shape. A rising graph should match increasing intervals, and a falling graph should match decreasing intervals.