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

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)?