When f'(x) changes from negative to positive, that point is a
When the derivative, f'(x), changes from negative to positive at a certain point on the graph of a function, it means that the function is experiencing a change in concavity
When the derivative, f'(x), changes from negative to positive at a certain point on the graph of a function, it means that the function is experiencing a change in concavity. Specifically, it indicates a change from concave downward to concave upward.
This point is called an inflection point. Inflection points are locations on the graph where the curvature changes. At an inflection point, the rate at which the function is increasing or decreasing also changes. In other words, the slope of the tangent line to the graph of the function changes at an inflection point.
To determine if a point where f'(x) changes from negative to positive is an inflection point, we need to investigate the behavior of the function and its second derivative, f”(x), around that point. If the second derivative changes sign at that point, then it is indeed an inflection point. If the second derivative does not change sign, it could be a point of interest but not necessarily an inflection point.
Note that not every point where f'(x) changes from negative to positive is an inflection point. It is possible for f'(x) to change signs at local extrema (maximum or minimum points) as well. The distinction lies in the change of concavity that occurs at an inflection point.
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