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_数学解答

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：443 ℃　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