How to Tell if a Function is Increasing or Decreasing by Calculus

To determine whether a function is increasing or decreasing using calculus, you should follow these steps:

1. Find the derivative: Start by computing the derivative of the function, denoted as f'(x). The derivative measures the rate of change of the function.
2. Set the derivative to zero: Solve the equation f'(x) = 0. The solutions will give you critical points where the function may change from increasing to decreasing or vice versa.
3. Test intervals: Choose test points in each of the intervals determined by the critical points. Substitute these test points into the derivative f'(x).
   • If f'(x) > 0 in an interval, then the function is increasing on that interval.
   • If f'(x) < 0 in an interval, then the function is decreasing on that interval.
4. Analyze the critical points: Evaluate the sign of the derivative just before and after each critical point to see how the function behaves around those points.

By following these steps, you can accurately determine where a function is increasing or decreasing. This analysis can provide valuable information about the behavior of the function and help you understand its overall shape.