| 单词 | extremum |
| 例句 | Find the local extrema for the function graphed in Exercise 1.28. Precalculus 2021-12-01T00:00:00Z Therefore, an extremum of the function is 1. Calculus, Volume 3 2016-03-30T00:00:00Z An absolute extremum may be positive, negative, or zero. Calculus, Volume 1 2016-03-30T00:00:00Z Find the local extrema for the function graphed in Exercise 3.28. Algebra and Trigonometry 2015-02-13T00:00:00Z It is really a universal trait, and it particularly manifests in times of extrema and adversity. Cory Doctorow on technological immortality, the transporter problem, and fast-moving futures 2017-07-16T04:00:00Z On reconsideration, however, I do not doubt that the received interpretation, which makes spatium extremum equivalent to finem, is the correct one. Notes and Queries, Number 211, November 12, 1853 A Medium of Inter-communication for Literary Men, Artists, Antiquaries, Genealogists, etc. Both of these values are greater than 1, leading us to believe the extremum is a minimum. Calculus, Volume 3 2016-03-30T00:00:00Z A function may have both an absolute maximum and an absolute minimum, just one extremum, or neither. Calculus, Volume 1 2016-03-30T00:00:00Z Together, local maxima and minima are called local extrema, or local extreme values, of the function. Algebra and Trigonometry 2015-02-13T00:00:00Z But if so, what is the extremum which Cicero advises Quintus to accept? The Letters of Cicero, Volume 1 The Whole Extant Correspodence in Chronological Order Later in this chapter we look at analytical methods for determining whether a function actually has a local extremum at a critical point. Calculus, Volume 1 2016-03-30T00:00:00Z To find extrema of functions of two variables, first find the critical points, then calculate the discriminant and apply the second derivative test. Calculus, Volume 3 2016-03-30T00:00:00Z Figure 4.13 shows several functions and some of the different possibilities regarding absolute extrema. Calculus, Volume 1 2016-03-30T00:00:00Z A function is also neither increasing nor decreasing at extrema. Algebra and Trigonometry 2015-02-13T00:00:00Z We will use graphical observations to determine whether a critical point is associated with a local extremum. Calculus, Volume 1 2016-03-30T00:00:00Z As shown in Figure 4.13, one or both of these absolute extrema could occur at an endpoint. Calculus, Volume 1 2016-03-30T00:00:00Z Can we find relative extrema of functions using derivatives? Calculus, Volume 3 2016-03-30T00:00:00Z Before looking at how to find absolute extrema, let’s examine the related concept of local extrema. Calculus, Volume 1 2016-03-30T00:00:00Z Note that we have to speak of local extrema, because any given local extremum as defined here is not necessarily the highest maximum or lowest minimum in the function’s entire domain. Algebra and Trigonometry 2015-02-13T00:00:00Z If an absolute extremum does not occur at an endpoint, however, it must occur at an interior point, in which case the absolute extremum is a local extremum. Calculus, Volume 1 2016-03-30T00:00:00Z Therefore, by Fermat’s Theorem, the point c at which the local extremum occurs must be a critical point. Calculus, Volume 1 2016-03-30T00:00:00Z The second derivative test for a function of one variable provides a method for determining whether an extremum occurs at a critical point of a function. Calculus, Volume 3 2016-03-30T00:00:00Z This idea is useful in determining where absolute extrema occur. Calculus, Volume 1 2016-03-30T00:00:00Z Figure 5 Definition of a local maximum These observations lead us to a formal definition of local extrema. Algebra and Trigonometry 2015-02-13T00:00:00Z With this idea in mind, let’s examine a procedure for locating absolute extrema. Calculus, Volume 1 2016-03-30T00:00:00Z At this point, we know how to locate absolute extrema for continuous functions over closed intervals. Calculus, Volume 1 2016-03-30T00:00:00Z Therefore the only possible values for the global extrema of f on D are the extreme values of f on the interior or boundary of D. Calculus, Volume 3 2016-03-30T00:00:00Z In these cases, how can we use a formula for a function to determine where these extrema occur? Calculus, Volume 1 2016-03-30T00:00:00Z TRY IT #4 Graph the function to estimate the local extrema of the function. Algebra and Trigonometry 2015-02-13T00:00:00Z We have also defined local extrema and determined that if a function f has a local extremum at a point c, then c must be a critical point of f . Calculus, Volume 1 2016-03-30T00:00:00Z However, c being a critical point is not a sufficient condition for f to have a local extremum at c. Calculus, Volume 1 2016-03-30T00:00:00Z The next step involves finding the extrema of f on the boundary of its domain. Calculus, Volume 3 2016-03-30T00:00:00Z The local extrema occur at x = 0, x = 1, and x = 2. Calculus, Volume 1 2016-03-30T00:00:00Z For the following exercises, use a graphing utility to estimate the local extrema of each function and to estimate the intervals on which the function is increasing and decreasing. Algebra and Trigonometry 2015-02-13T00:00:00Z Earlier in this chapter we stated that if a function f has a local extremum at a point c, then c must be a critical point of f . Calculus, Volume 1 2016-03-30T00:00:00Z However, a function is not guaranteed to have a local extremum at a critical point. Calculus, Volume 1 2016-03-30T00:00:00Z The next step involves finding the extrema of g on the boundary of its domain. Calculus, Volume 3 2016-03-30T00:00:00Z As mentioned earlier, if f has a local extremum at a point x = c, then c must be a critical point of f . Calculus, Volume 1 2016-03-30T00:00:00Z We can find local extrema from a graph. Algebra and Trigonometry 2015-02-13T00:00:00Z Note that f need not have a local extrema at a critical point. Calculus, Volume 1 2016-03-30T00:00:00Z The critical points are candidates for local extrema only. Calculus, Volume 1 2016-03-30T00:00:00Z These are the absolute extrema of g on D as shown in the following figure. Calculus, Volume 3 2016-03-30T00:00:00Z Since f has a local extremum at c, f has a local maximum or local minimum at c. Calculus, Volume 1 2016-03-30T00:00:00Z Even then, finding where extrema occur can still be algebraically challenging. Algebra and Trigonometry 2015-02-13T00:00:00Z In Figure 4.31, we show that if a continuous function f has a local extremum, it must occur at a critical point, but a function may not have a local extremum at a critical point. Calculus, Volume 1 2016-03-30T00:00:00Z We show that if f has a local extremum at a critical point, then the sign of f ′ switches as x increases through that point. Calculus, Volume 1 2016-03-30T00:00:00Z The function f does not have a local extremum at c. Calculus, Volume 1 2016-03-30T00:00:00Z In other words, local extrema can only occur at critical points. Calculus, Volume 1 2016-03-30T00:00:00Z The sign of f ′ changes at all local extrema. Calculus, Volume 1 2016-03-30T00:00:00Z Using Figure 4.31, we summarize the main results regarding local extrema. Calculus, Volume 1 2016-03-30T00:00:00Z If a continuous function f has a local extremum, it must occur at a critical point c. Calculus, Volume 1 2016-03-30T00:00:00Z We can summarize the first derivative test as a strategy for locating local extrema. Calculus, Volume 1 2016-03-30T00:00:00Z Note this theorem does not claim that a function f must have a local extremum at a critical point. Calculus, Volume 1 2016-03-30T00:00:00Z Now let’s look at how to use this strategy to locate all local extrema for particular functions. Calculus, Volume 1 2016-03-30T00:00:00Z We know that if a continuous function has a local extrema, it must occur at a critical point. Calculus, Volume 1 2016-03-30T00:00:00Z However, a function need not have a local extrema at a critical point. Calculus, Volume 1 2016-03-30T00:00:00Z Here we examine how the second derivative test can be used to determine whether a function has a local extremum at a critical point. Calculus, Volume 1 2016-03-30T00:00:00Z Rather, it states that critical points are candidates for local extrema. Calculus, Volume 1 2016-03-30T00:00:00Z Is it possible for a point c to be both an inflection point and a local extrema of a twice differentiable function? Calculus, Volume 1 2016-03-30T00:00:00Z Consequently, by the extreme value theorem, we were guaranteed that the functions had absolute extrema. Calculus, Volume 1 2016-03-30T00:00:00Z Many functions still have at least one absolute extrema, even if the domain is not closed or the domain is unbounded. Calculus, Volume 1 2016-03-30T00:00:00Z Therefore, we can still consider functions over unbounded domains or open intervals and determine whether they have any absolute extrema. Calculus, Volume 1 2016-03-30T00:00:00Z In some of these cases, the functions have local extrema at critical points, whereas in other cases the functions do not. Calculus, Volume 1 2016-03-30T00:00:00Z A function may have both an absolute maximum and an absolute minimum, have just one absolute extremum, or have no absolute maximum or absolute minimum. Calculus, Volume 1 2016-03-30T00:00:00Z If a function has a local extremum, the point at which it occurs must be a critical point. Calculus, Volume 1 2016-03-30T00:00:00Z However, a function need not have a local extremum at a critical point. Calculus, Volume 1 2016-03-30T00:00:00Z Each extremum occurs at a critical point or an endpoint. Calculus, Volume 1 2016-03-30T00:00:00Z A function may or may not have a local extremum at a critical point. Calculus, Volume 1 2016-03-30T00:00:00Z Look for critical points to locate local extrema. Calculus, Volume 1 2016-03-30T00:00:00Z |
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