If f(x) is a function , then, if for some x=M there exitsts an an f(M) such that f(x) < f(M), for all x ,them f(M) is the absolute maximum.
Similarly if there exists an f(m) such that f(m) < f(x) for all x, then f(m) is the absolute minimum.
The local maximums are the values of function where the slope of the functions changes from positive to negative or from negative to positive .So the value of the slope has to to be zero.
An extreme point of a function is found by differentiating the expression of the function and, after that, solving the equation of the differentiated function.
In the point where the first derivative is cancelling, the function has a minimum or maximum point.
We know that maximum or minimum points could be either relative, or absolute. The relative maximum or minimum is also called the local maximum or minimum. That means that the tangent to the graph of function in the point where the first derivative is cancelling, is horizontal but is not the highest or lowest value of the function.
An absolute maximum or minimum is the point where the tangent is horizontal and there is no other value of the function higer or lower than this one.
Note that the relative maximum or minimum could be an absolute maximum or minimum.