When f ‘(x) changes fro positive to negative, f(x) has a
relative maximum
local maximum at x.
This statement follows from the definition of the derivative as the instantaneous rate of change of a function. If the derivative f ‘(x) changes from positive to negative at point x, it means that the slope of the tangent line to the graph of the function at this point changes from increasing to decreasing. That is, the function is changing from having a positive slope to having a negative slope. This implies that the graph of the function is reaching a peak, or a local maximum, at point x.
To visualize this, imagine the graph of the function as a roller coaster ride. As you go up the first hill, the slope of the track is increasing, and you feel like you are accelerating. However, at the top of the hill, the slope of the track begins to decrease, and you feel like you are decelerating. This is the point of maximum height on the roller coaster track, or the local maximum of the function.
In summary, when f ‘(x) changes from positive to negative, it means that the function is reaching a local maximum at point x. This knowledge can be very useful in optimization problems, where we want to find the maximum or minimum value of a function over a given interval or domain.
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