When f ‘(x) changes from negative to positive, f(x) has a
relative minimum
local minimum.
When the first derivative of a function, f'(x), changes from negative to positive at a specific point, it means that the slope of the tangent line to the graph of f(x) is changing from negative (sloping downward) to positive (sloping upward) at that point. This indicates a change from a decreasing trend to an increasing trend in the function.
Since the tangent line represents the instantaneous rate of change of the function at that point, a change in slope indicates a change in the rate of change. More specifically, it indicates a transition from a negative rate of change (where the function is getting smaller) to a positive rate of change (where the function is increasing).
This change in the rate of change occurs at a local minimum because at this point the slope of the function changes from decreasing to increasing. At a local minimum, the function has a lower value than all nearby points, forming a valley in the graph. This is because the function is decreasing before the minimum point and increasing after it. The minimum point represents the bottom of the valley and is the point where the slope changes from negative to positive.
Therefore, when f ‘(x) changes from negative to positive, f(x) has a local minimum.
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