How do relative (or local) minimum points, relative (or local)

How do relative (or local) minimum points, relative (or local) maximum points, and inflection points affect the shape of the graph of a given function?  That is, why might someone use derivative techniques to find these points before sketching the graph of a function (assuming no graphing calculator)?

How does someone determine the intervals over which a function is increasing or decreasing?  What has to be true about the slope of the tangent in each situation (increasing vs. decreasing)?

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