# 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 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.

• 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.