121616949-math.119

# 121616949-math.119 - a and b and we want to know the...

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6 Applications of the Derivative Many important applied problems involve finding the best way to accomplish some task. Often this involves finding the maximum or minimum value of some function: the minimum time to make a certain journey, the minimum cost for doing a task, the maximum power that can be generated by a device, and so on. Many of these problems can be solved by finding the appropriate function and then using techniques of calculus to find the maximum or the minimum value required. Generally such a problem will have the following mathematical form: Find the largest (or smallest) value of f ( x ) when a x b . Sometimes a or b are infinite, but frequently the real world imposes some constraint on the values that x may have. Such a problem differs in two ways from the local maximum and minimum problems we encountered when graphing functions: We are interested only in the function between
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Unformatted text preview: a and b , and we want to know the largest or smallest value that f ( x ) takes on, not merely values that are the largest or smallest in a small interval. That is, we seek not a local maximum or minimum but a global maximum or minimum, sometimes also called an absolute maximum or minimum. Any global maximum or minimum must of course be a local maximum or minimum. If we ﬁnd all possible local extrema, then the global maximum, if it exists , must be the largest of the local maxima and the global minimum, if it exists , must be the smallest of the local minima. We already know where local extrema can occur: only at those points at which f ± ( x ) is zero or undeﬁned. Actually, there are two additional points at which a 105...
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