In calculus, the concepts of absolute minimum and local minimum refer to specific points within the graph of a function where certain conditions regarding values are satisfied.
An absolute minimum of a function is the lowest point over the entire domain of the function. This means that the value of the function at this point is less than or equal to the value of the function at any other point in that domain. If a function has an absolute minimum at a point x = c, it means that for all x in the domain of the function, f(c) ≤ f(x).
On the other hand, a local minimum refers to a point where the function value is lower than the values of the function in a neighboring region, but not necessarily the entire domain. At a local minimum, there exists some interval around x = c such that for all x within that interval, f(c) < f(x).
To summarize, the key difference is that an absolute minimum is the lowest point across the whole function, while a local minimum is only the lowest point within a small surrounding area. A function can have multiple local minimums but only one absolute minimum.