5.3 Determining Intervals on Which a Function Is Increasing or Decreasing

A function is increasing on an interval if its values rise as x increases, and it is decreasing on an interval if its values fall as x increases. Derivatives provide a direct way to determine this behavior: if f'(x)>0 on an interval, then f is increasing there, and if f'(x)<0 on an interval, then f is decreasing there. So the sign of the derivative tells us the direction of the function’s movement. This is one of the most important interpretations of the derivative in graph analysis.