Section 4.5: Absolute Maxima and Minima

Absolute Maxima and Minima

Extreme Value Theorem

A function f that is continuous on a closed interval [a, b] has both an absolute maximum and an absolute minimum on that interval.

Locating Absolute Extrema

Absolute extrema (if they exist) must occur at critical numbers or at endpoints.

Finding Absolute Extrema on a Closed Interval

  1. Check to make certain that f is continuous over [a, b].
  2. Find the critical numbers in the interval (a, b).
  3. Evaluate f at the endpoints a and b and at the critical numbers found in step 2.
  4. The absolute maximum of f on [a, b] is the largest value found in step 3.
  5. The absolute minimum of f on [a, b] is the smallest value found in step 3.

Second-Derivative Test for Local Extrema

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 upwardLocal minimum
0-Concave downwardLocal maximum
00?Test does not apply

Second-Derivative Test for Absolute Extrema on an Open Interval

Let f be continuous on an open interval I with only one critical number c in I.