Section 4.5: Absolute Maxima and Minima

Absolute Maxima and Minima

• If f(c) ≥ f(x) for all x in the domain of f, then f(c) is called the absolute maximum of f.
• If f(c) ≤ f(x) for all x in the domain of f, then f(c) is called the absolute minimum of f.
• An absolute maximum or absolute minimum is called an absolute extremum.

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 upward	Local minimum
0	-	Concave downward	Local maximum
0	0	?	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.

• If f'(c) = 0 and f''(c) > 0, then f(c) is the absolute minimum of f on I.
• If f'(c) = 0 and f''(c) < 0, then f(c) is the absolute maximum of f on I.