When f ‘(x) changes from positive to negative, f(x) has a
local maximum
local maximum.
In calculus, a “local maximum” refers to a point on a graph where the function value reaches a peak within a small interval. When the derivative of a function f ‘(x) changes from positive to negative at a specific point x, it indicates that the slope of the function is initially increasing and then decreases, signifying a change from an upward trend to a downward trend at that point.
In simpler terms, if f(x) represents a function, and its derivative f ‘(x) transitions from positive to negative at a specific x-value, it means that the function is initially rising but then starts to fall at that point. Therefore, the function value f(x) has a local maximum there.
For example, consider the function f(x) = x^3 – 3x^2 – 9x + 5. If we find the derivative f ‘(x), we get f ‘(x) = 3x^2 – 6x – 9. To determine where f(x) has a local maximum, we need to find the points where f ‘(x) changes sign. Solving the equation f ‘(x) = 0, we find that x = -1 and x = 3. By analyzing the sign of f ‘(x) in the intervals (-∞, -1), (-1, 3), and (3, +∞), we find that f ‘(x) changes from positive to negative at x = 3. Therefore, f(x) has a local maximum at x = 3.
To summarize, when f ‘(x) changes from positive to negative, it means that the function is transitioning from an increasing slope to a decreasing slope, indicating a local maximum for f(x) at that specific point.
More Answers:
Understanding Increasing Behavior | The Significance of a Positive Derivative for Function f(x)Understanding the concept of a negative derivative | Decreasing Functions and Negative Slope
The Significance of Changing Derivative | Transition from Decreasing to Increasing Behavior