When f'(x) changes from positive to negative, that turning point is a
When the derivative of a function, f'(x), changes from positive to negative, it indicates that the function’s graph is transitioning from increasing to decreasing
When the derivative of a function, f'(x), changes from positive to negative, it indicates that the function’s graph is transitioning from increasing to decreasing. This turning point is known as a local maximum or a relative maximum.
A local maximum is a point on a function where the value of the function is larger than the values of the function at nearby points, within a specific interval. In other words, it represents a peak or a highest point in that specific region of the function.
To better understand this concept, let’s consider an example. Suppose we have a function f(x) and take its derivative, f'(x). If the derivative changes from positive to negative at a specific value of x, say x = a, it means that to the left of x = a, the slope of the function is positive (indicating an increasing behavior), and to the right of x = a, the slope becomes negative (indicating a decreasing behavior).
At this turning point, the value of the function itself at x = a would represent the local maximum. This means that f(a) is the highest value that can be attained within a specific interval around x = a.
In summary, when f'(x) changes from positive to negative, it signifies a transition from increasing to decreasing behavior of the function, and the corresponding turning point is a local maximum or a relative maximum.
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