When f ‘(x) changes fro positive to negative, f(x) has a
local maximum
local maximum.
In calculus, the derivative f ‘(x) represents the rate of change of a function f(x) at any given point x. When f ‘(x) changes from positive to negative, this means that the slope of the function is initially increasing but then starts decreasing.
When the derivative changes sign from positive to negative, it indicates that the function is changing from increasing to decreasing. In other words, the function is reaching a peak or maximum value at that particular point.
Hence, when f ‘(x) changes from positive to negative, we can conclude that f(x) has a local maximum at that point. A local maximum is a point on the graph of the function where the function reaches a high point, but it may not be the absolute highest point on the entire function.
More Answers:
Understanding the Concept of Positive Derivatives: Analyzing the Increasing Behavior of FunctionsUnderstanding the Significance of a Negative f ‘(x): Rate of Change and Function Behavior Explained
Understanding the Change in Concavity of a Function: Exploring Inflection Points and Rate of Change