The Relationship Between Positive Derivative and Local Minimum Points in Functions

When f ‘(x) changes from negative to positive, f(x) has a

When the derivative of a function f'(x) changes from negative to positive, it indicates that the function f(x) is going through a local minimum point

When the derivative of a function f'(x) changes from negative to positive, it indicates that the function f(x) is going through a local minimum point.

To understand why this is the case, let’s break it down step by step:

1. The derivative f'(x) of a function f(x) represents the rate of change of the function at any given point. If f'(x) is negative, it means that the function is decreasing at that point, while if f'(x) is positive, it means that the function is increasing.

2. When f'(x) changes from negative to positive, it means that the function is transitioning from a decreasing slope to an increasing slope. This transition occurs at a specific point, which is where the derivative changes from negative to positive.

3. At this point, the slope of f(x) changes from negative to zero, and then becomes positive as x increases. This means that the function is going through a local minimum. A local minimum is a point where the function reaches its lowest value within a specific interval.

4. In graphical terms, if we were to plot the function f(x) on a graph, the local minimum would correspond to the lowest point on the curve within the interval where f'(x) changes from negative to positive. The curve would be sloping downwards before reaching this point, and then start sloping upwards after it.

It’s important to note that a local minimum is not necessarily the absolute minimum of the function. There could be other points where the slope changes from positive to negative, indicating the presence of other local minimums or maximums. To determine if a local minimum is also the absolute minimum, further analysis or information about the function is usually required.

To summarize, when f ‘(x) changes from negative to positive, f(x) has a local minimum. This means that the function is transitioning from a decreasing slope to an increasing slope at a specific point.

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