what is the difference between the local min and max and the absolute min and max?
"local minimum" must be a lower value than any values of the function in some small interval around the point- but can be larger than other values.
For example, has a local maximum at x= 0 and a local minimum at x= 1. f(0)= 1 while values of x < 0 or x< about 1.2 but then the values become higher. Similarly, for values of x close to 1, f(x) is larger than f(1) but for x< -1, f(x) is less than f(1).
An "absolute maximum" (also called "global maximum") must be larger than or equal to any value of the function- on what ever set it is the global maximum for. Similarly, a "local minimum" (also called "global minimum" must be less than or equal to any value f(x) for any x in the set.
The example does NOT have a global max or min for all real values of x. If we restrict to , say x= -2 to x= 2, then f(-2)= -8- 4+ 2+ 1= -9 is an absolute minimum while f(2)= 8- 4- 2+ 1= 3 is an absolute maximum.