AP Calculus Review Finding Absolute Extrema

The absolute extrema of a function are the largest and smallest values of the function. What is the most profit that a company can make? What is the least amount of fence needed to enclose a garden? Once you know how to find the absolute extrema of a function, then you can answer these kinds of questions and many more! Overview: What are Absolute Extrema? The absolute extrema of a function f on a given domain set D are the absolute maximum and absolute minimum values of f(x) as x ranges throughout D. In other words, we say that M is the absolute maximum if M = f(c) for some c in D, and f(x) M for all other x in D. We define the absolute minimum m in much the same way, except that f(x) m for all x in D. Functions with Discontinuity Sometimes a function may fail to have an absolute minimum or maximum on a given domain set. This often happens when the function has a discontinuity. This function is discontinuous on the interval shown. It has an absolute minimum value, 0, but no absolute maximum. Domain Sets and Extrema Even if the function is continuous on the domain set D, there may be no extrema if D is not closed or bounded. For example, the parabola function, f(x) = x2 has no absolute maximum on the domain set (-, ). This is because the values of x2 keep getting larger and larger without bound as x . By the way, this function does have an absolute minimum value on the interval: 0. However, there may still be issues even on a bounded domain set. The function below has neither absolute minimum nor maximum because the endpoints of the interval are not in its domain. Note, the open circles on the graph mean that those points are missing, so there cannot be any extrema at those points. This graph is defined on the open interval, (- The absolute maximum value is 39 (at x = -2), and the absolute minimum is -42 (at x = -5). Looking at the graph of f, you can verify the max and min values. Summary The absolute extrema of a function on a given domain set D are the greatest and least values of the function on D. The Extreme Value Theorem guarantees that a continuous function must have absolute extrema on a bounded, closed interval. You can use the Closed Interval Method to locate the absolute extrema. Now that you know more about absolute extrema, you can maximize your score on the AP Calculus exams!