{"id":9220,"date":"2017-03-03T18:19:38","date_gmt":"2017-03-04T02:19:38","guid":{"rendered":"https:\/\/magoosh.com\/hs\/?p=9220"},"modified":"2017-03-03T18:20:05","modified_gmt":"2017-03-04T02:20:05","slug":"ap-calculus-review-finding-absolute-extrema","status":"publish","type":"post","link":"https:\/\/magoosh.com\/hs\/ap\/ap-calculus-review-finding-absolute-extrema\/","title":{"rendered":"AP Calculus Review: Finding Absolute Extrema"},"content":{"rendered":"

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!<\/p>\n

Overview: What are Absolute Extrema?<\/h2>\n

The absolute extrema<\/strong> of a function f<\/em> on a given domain set D<\/em> are the absolute maximum and absolute minimum values of f<\/em>(x<\/em>) as x<\/em> ranges throughout D<\/em>. <\/p>\n

In other words, we say that M<\/em> is the absolute maximum if M<\/em> = f<\/em>(c<\/em>) for some c<\/em> in D<\/em>, and f<\/em>(x<\/em>) ≤ M<\/em> for all other x<\/em> in D<\/em>.<\/p>\n

We define the absolute minimum m<\/em> in much the same way, except that f<\/em>(x<\/em>) ≥ m<\/em> for all x<\/em> in D<\/em>.<\/p>\n

\"Graph<\/p>\n

Functions with Discontinuity<\/h3>\n

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.<\/p>\n

\"Graph
This function is discontinuous on the interval shown. It has an absolute minimum value, 0, but no absolute maximum.<\/figcaption><\/figure>\n

Domain Sets and Extrema<\/h3>\n

Even if the function is continuous on the domain set D<\/em>, there may be no extrema if D<\/em> is not closed<\/em> or bounded<\/em>. <\/p>\n

For example, the parabola function, f<\/em>(x<\/em>) = x<\/em>2<\/sup> has no absolute maximum on the domain set (-∞, ∞). This is because the values of x<\/em>2<\/sup> keep getting larger and larger without bound as x<\/em> → ∞. By the way, this function does have an absolute minimum value on the interval: 0.<\/p>\n

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.<\/p>\n

\"Graph
This graph is defined on the open interval, (-4, 4). There are no absolute extrema.<\/figcaption><\/figure>\n

The Extreme Value Theorem<\/h3>\n

In practice, we usually require D<\/em> to be a closed interval of the form [a<\/em>, b<\/em>] for some constants a<\/em> < b<\/em>. In that case, the Extreme Value Theorem<\/strong> guarantees both absolute extrema must<\/em> exist.<\/p>\n