A function f that is continuous on a closed interval [a, b] has both an absolute maximum and an absolute minimum on that interval.
Absolute extrema (if they exist) must occur at critical numbers or at endpoints.
Let c be a critical number of f(x) such that f'(c) = 0. If the second derivative f''(c) > 0, then f(c) is a local minimum. If f''(c) < 0, then f(c) is a local maximum.
|f'(c)||f''(c)||Graph of f is:||f(c)|
|0||+||Concave upward||Local minimum|
|0||-||Concave downward||Local maximum|
|0||0||?||Test does not apply|
Let f be continuous on an open interval I with only one critical number c in I.