Maxima & Minima
Absolute (Global) Extreme Values
Introduction: One of the most useful things we can learn from a function's derivative is whether the function assumes any maximum or minimum values on a given interval and where these values are located if it does. Once we know how to find a function's extreme values, we will be able to answer such questions as "What is the maximum acceleration of a space shuttle?" and "What is the radius of a contracted windpipe that expels air most rapidly during a cough?".
Absolute (Global) extreme values: A function f has an absolute or global maximum at 'c' if f(c) ≥ f(x) for all 'x' in domain of the function f. The number f(c) is called the absolute maximum value of f on its domain. Similarly, f has an absolute or global minimum at 'c' if f(c) ≤ f(x) for all 'x' in domain of the function f. The number f(c) is called the absolute minimum value of f on its domain.
Together, the absolute minimum and the absolute maximum are known as the absolute (global) extrema of the function. We often skip the term absolute or global and just say maximum and minimum values of the function.
Following figure shows the graph of a function f with absolute maximum at 'c' and absolute minimum at 'a'. The value of f at 'a', that is, f(a) is called the absolute minimum value and the value of f at 'c', that is, f(c) is called the absolute maximum value of the function f.
The extreme value theorem: If the function f is continuous on a closed interval [a, b], then f attains the maximum value f(c) and the minimum value f(d) at some numbers 'c' and 'd' in the interval [a, b].
|Closed interval method
|To find an absolute extrema of a continuous function f on a closed interval [a, b]:
- Find the values of f at the critical numbers [a number in the interior of the domain of a function f(x) at which = 0 or does not exist] of f in (a, b).
- Find the values of f at the endpoints of the interval.
- The largest of the values from above two steps is the absolute maximum value of f; the smallest of these values is the absolute minimum value of f.