When f ‘(x) changes from positive to negative, f(x) has a
When f'(x) changes from positive to negative, it indicates that the slope of the function f(x) is decreasing
When f'(x) changes from positive to negative, it indicates that the slope of the function f(x) is decreasing. This means that the rate at which the function is increasing is slowing down and eventually starts decreasing.
In terms of the behavior of f(x), when f'(x) changes from positive to negative at a specific value of x, it implies that the function f(x) reaches a local maximum at that point. A local maximum is a point on the graph of the function where the function reaches its highest value in a local interval.
To understand this concept, let’s take an example. Consider a simple quadratic function, f(x) = x^2. The derivative of this function is f'(x) = 2x.
When x is negative, f'(x) is negative, indicating that the slope of f(x) is decreasing. As x approaches 0 from the left side, f'(x) approaches 0, meaning the slope is flattening out. Then, as x becomes positive, f'(x) becomes positive, demonstrating that the slope of f(x) is increasing.
In this case, f'(x) changes from positive to negative at x = 0. At this point, f(x) reaches a local maximum, which is 0, because the function changes from increasing to decreasing around this point. After x = 0, f(x) starts to decrease.
So, when f'(x) changes from positive to negative, f(x) has a local maximum.
More Answers:
Understanding Positive Derivatives | When a Function’s Derivative is Positive, It Means the Function is IncreasingUnderstanding the Trend of a Function | Decreasing Behavior and Negative Derivatives
Understanding the Significance of f'(x) Changing from Negative to Positive | Exploring the Behavior and Properties of Mathematical Functions