The graph of a derivative f^' is shown. Use the information in each graph to determine where f i

Step 1: The function $f$ is increasing where its derivative $f'$ is positive and decreasing wher

The graph of a derivative f^' is shown. Use the information...

The graph of a derivative $f^{\prime}$ is shown. Use the information in each graph to determine where f is increasing or decreasing and the $x$ -values of any extrema. Then sketch a possible graph of $f$
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Key Concepts

Relationship between f and f'

Understanding the connection between a function f and its derivative f' is fundamental. The derivative represents the slope or rate of change of the function. When f' is positive, the original function is increasing; when f' is negative, the function is decreasing. The zeros of f' indicate points where the slope is zero, which are potential locations for local maxima or minima on f.

Increasing and Decreasing Functions

A key concept is that the sign of the derivative determines where a function is increasing or decreasing. If f'(x) > 0 for an interval, then f is increasing on that interval; if f'(x) < 0, f is decreasing. This information is used to identify intervals of monotonicity in the original function based solely on the graph or behavior of its derivative.

Critical Points and Extrema

Critical points occur where the derivative equals zero or does not exist, which are candidates for local extrema. By examining the sign changes of the derivative around these points, one can apply the First Derivative Test. A change in f' from positive to negative indicates a local maximum, while a change from negative to positive indicates a local minimum. This analysis is essential for understanding the overall shape of the function f.

Sketching f from f'

Another important concept is how to reconstruct a potential graph of the original function using the graph of its derivative. By integrating the information about where f increases or decreases and noting the locations of the critical points, one can sketch the behavior of f. While the exact function might not be determined, the general shape—rising or falling in certain intervals with appropriate turning points—can be deduced.

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