Part 2/7:
To begin, let's define local maximums and minimums. A local maximum at a certain point ( C ) on a function ( f(x) ) is the highest value of ( f(x) ) within some surrounding range, while a local minimum at point ( C ) is the lowest value within that same vicinity. In simple terms, if you were to draw a graph, a local maximum looks like a peak, whereas a local minimum resembles a trough.
Example Graphs
Consider the curve of a function in the region from ( a ) to ( b ). For a function illustrated as shown, if ( C ) represents a local maximum, then values surrounding ( C ) are less than ( f(C) ). Conversely, a point ( D ) where the value is lower than nearby points designates a local minimum.