When f ‘(x) changes from positive to negative, f(x) has a
When f ‘(x) changes from positive to negative, it means that the derivative of the function f(x) changes from positive to negative at that particular point
When f ‘(x) changes from positive to negative, it means that the derivative of the function f(x) changes from positive to negative at that particular point. This implies that the function f(x) is decreasing at that point.
In other words, if f ‘(x) is positive for values of x to the left of this point, and then becomes negative for values of x to the right of this point, it suggests that the slope of the tangent line to the graph of f(x) is initially positive and then becomes negative as x increases.
This change in the sign of the derivative indicates that f(x) has reached a local maximum at that point. A local maximum is a point on the graph of f(x) where the function reaches its highest value in a specific interval around that point.
To visualize this, imagine a hill or a peak on a graph. As you climb up the hill or approach the peak, you initially encounter an upward slope (positive derivative), but as you start descending from the top, you experience a downward slope (negative derivative).
Therefore, when f ‘(x) changes from positive to negative, f(x) has a local maximum at that point.
More Answers:
Understanding the Positive Derivative: How it Indicates an Increasing FunctionUnderstanding Decreasing Functions: Analyzing the Negative Derivative and Behavior of Functions
Understanding Changing Trends in Functions: The Significance of f'(x) Changing from Negative to Positive