f'(x) changes from positive to negative
f(x) has a relative maximum
If f'(x) changes from positive to negative, it means that the slope of the graph of f(x) is first increasing and then decreasing. This change in slope can occur at a critical point or at an inflection point.
If the change in slope occurs at a critical point, it means that the function has a local maximum at that point. The derivative is positive to the left of the critical point and negative to the right of the critical point. This means that the slope of the graph is increasing on the left side of the critical point and decreasing on the right side of the critical point. The critical point is where the slope of the graph changes from positive to negative.
If the change in slope occurs at an inflection point, it means that the concavity of the graph changes from upward to downward. The graph initially curves upward, which means that the derivative is increasing, and then it curves downward, which means that the derivative is decreasing. The inflection point is where the second derivative of the function changes sign.
In either case, the change from positive to negative slope indicates a change in the behavior of the function. It could be a maximum or a minimum, or a change in the curvature of the graph. Understanding the behavior of the function in this way can be useful in optimization problems, where we need to find maximum or minimum values of a function.
More Answers:
How the Second Derivative Can Help Find Inflection Points, Sketch Accurate Graphs, and Determine Maximum or Minimum Values of a Functionthe Relationship between Concavity and the Second Derivative
the Significance of f'(x) Changing from Negative to Positive in Math: Exploring Critical Points and Turning Points.