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Calculus AB: Applications of the Derivative

Absolute and Local Extrema

Problems for "Related Rates"

Absolute and Local Extrema, page 2

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Figure %: Graph of function f on the interval [a, e]

Absolute Extrema

If a function has an absolute maximum at x = b , then f (b) is the largest value that f can attain. Similarly, if a function has an absolute minimum at x = b , then f (b) is the smallest value that f can attain.

On the graph above of the function f on the closed interval [a, e] , the point (a, f (a)) represents the absolute minimum, and the point (d, f (d )) represents the absolute maximum.

To define these terms more formally: a function f has an absolute maximum at x = b if f (b)≥f (x) for all x in the domain of f . A function f has an absolute minimum at x = b if f (b)≤f (x) for all x in the domain of f . Together, the absolute minimum and the absolute maximum are known as the absolute extrema of the function.

Local Extrema

A function f has a local maximum at x = b if f (b) is the largest value that f attains "near b ." Similarly, a function f has a local minimum at x = b if f (b) is the smallest value that f attains "near b ." In the graph above, f has a local maximum at x = b and at x = d , and has a local minimum at x = c .

Taken together, the local maxima and local minima are known as the local extrema. A local minimum or local maximum may also be called a relative minumum or relative maximum.

To define these terms more formally: a function f (x) has a local (or relative) maximum at x = b if there is an open interval I in which f (b)≥f (x) for all x in I . A function f (x) has a local (or relative) minimum at x = b if there is an open interval I in which f (b)≤f (x) for all x in I .

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