Use the given graph of y=f^'(x) to find the intervals on which f is increasing, the intervals on

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Use the given graph of y=f^'(x) to find the intervals on...

Use the given graph of $y=f^{\prime}(x)$ to find the intervals on which $f$ is increasing, the intervals on which $f$ is decreasing, and the x coordinates of the local extrema off. Sketch a possible graph of $y=f(x)$.

Key Concepts

Increasing and Decreasing Intervals

Identifying the intervals where a function is increasing or decreasing involves analyzing the sign of its derivative. When f?(x) is greater than zero, the function is increasing; when f?(x) is less than zero, the function is decreasing. This concept helps in understanding the overall behavior of the function.

Critical Points

Critical points occur where the derivative is zero or undefined. These points are key to determining the potential locations of local extrema as they mark where the function's rate of change may change from positive to negative or vice versa.

Graphical Analysis of Derivative Information

Using the graph of the derivative to infer the behavior of the original function involves observing the sign and value changes of the derivative. By identifying regions where the derivative graph is above or below the x-axis and noting any changes in its trend, one can sketch a credible graph of the original function and determine where local maxima and minima occur.

First Derivative Test

The First Derivative Test is used to determine local maximum and minimum points of a function by examining changes in the sign of its derivative. If the derivative changes from positive to negative at a critical point, the function has a local maximum; if it changes from negative to positive, the function has a local minimum.

Relationship Between a Function and Its Derivative

A function's derivative gives the rate of change or slope of the function at any point. When the derivative is positive over an interval, the function is increasing on that interval; when the derivative is negative, the function is decreasing. This relationship is fundamental in analyzing the behavior of functions.

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