The Mean Value Theorem and Its Consequences

The point (M,f(M)) is called an absolute maximum of f if f(x) ≤ f(M) for every x in the domain of f . The point (m, f(m)) is called an absolute minimum of f if f(x) ≥ f(m) for every x in the domain of f . More than one absolute maximum or minimum may exist. For example, if f(x) = |x| for x ∈ [−1, 1] then f(x) ≤ 1 and there are absolute maxima at (1, 1) and at (−1, 1), but only one absolute minimum, at (0, 0). Recall that if f is a continuous function with domain [a, b] then there is some M ∈ [a, b] such that f(x) ≤ f(M) for all x ∈ [a, b] and there is some m ∈ [a, b] such that f(x) ≥ f(m). In fact, what we know is that the range of f is [f(m), f(M)]. We want to see what additional information the differentiability of f gives us.