Understanding the Significance of a Change in Slope | From Decreasing to Increasing Rate of Change

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 at a specific point, it indicates a change in the behavior of the original function, f(x), at that point

When the derivative of a function, f ‘(x), changes from negative to positive at a specific point, it indicates a change in the behavior of the original function, f(x), at that point. Specifically, it signifies a change from a decreasing rate of change to an increasing rate of change.

In terms of the graph of the function, this means that as you move from left to right on the x-axis, the function initially decreases, then reaches a point where it starts increasing. This occurs at the specific x-value where the derivative changes sign, from negative to positive.

The point where f ‘(x) changes sign is called a critical point or a turning point. At this point, the instantaneous slope of the function changes from negative (meaning the function is decreasing) to positive (meaning the function is increasing).

It’s important to note that when f ‘(x) changes from negative to positive, it does not guarantee that f(x) is increasing continuously beyond that point. There might still be regions of the function where it is decreasing or has other behaviors. However, at the specific point of the sign change, f(x) experiences a change in slope from negative to positive.

To summarize, when f ‘(x) changes from negative to positive, f(x) has a turning point or critical point where the function changes from decreasing to increasing slope.

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