L17 - Lecture 17 Critical Numbers Extreme Values One of the...

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Lecture 17 — Critical Numbers, Extreme Values One of the most important applications of calculus is finding the extremes of functions. For example, we may be interested in finding the least cost, the shortest distance, or the largest possible area within the context of our world. In order to find a largest or smallest value of a function, we need to explore the questions: (1) What defining characteristic are we looking for? (2) When does one exist? or is our search in vain? (3) Where should we look for them? These questions will now be answered through a series of simple definitions and theorems to develop a strategy to find the extrema of a function. Definition: Function f ( x ) is said to have an absolute maximum (minimum) at c if Absolute maxima and minima are collectively called
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Note that by its definition, an absolute extremum must be a value that the function attains (an actual f ( x ) output value). Examine the graphs of the function f ( x ) = 1 x below.
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