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已知函数f﹙x﹚=tanx .
﹙1﹚试分析f﹙x﹚在区间﹙0,2π﹚上的单调性,并求函数y=f﹙2x﹚的图像的对称中心;
﹙2﹚已知α,β∈﹙π/2,π﹚且tanα<cotβ,试根据f﹙x﹚的单调性证明:α+β<3π/2.
人气:406 ℃ 时间:2020-02-04 07:57:33
解答
(1) when x = pi/2 or 3pi/2 tan(x) = infinity
(0,pi/2) (pi/2,3pi/2),(3pi/2,2pi) are three intervals,f(x) increases
f(x) is symmetric about npi
so f(2x) is symmetric about npi/2
(2) when A and B are in (pi/2,pi),tanA < 0 and tanB < 0
tan A < 1/tanB
tanA*tanB > 1
tan(A+B) = (tanA+tanB)/(1-tanAtanB) > 0
therefore A+B must be between pi and 3pi/2
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