［2sin2x﹡(1-tanx)ˆ2］/［1+(tanx)ˆ2］的单调区间?

Question

［2sin2x﹡(1-tanx)ˆ2］/［1+(tanx)ˆ2］的单调区间?

Answer 1

设f(x)=［2sin2x﹡(1-tanx)ˆ2］/［1+(tanx)ˆ2］=[2sin2x﹡(1-2tanx+(tanx)^2)]/[1+(tanx)^2]
=[2sin2x﹡(-2tanx)]/[1+(tanx)^2]+2sin2x=-4sin2x﹡tanx/(secx)^2+2sin2x
=-4sin2x﹡tanx﹡(cosx)^2+2sin2x=-2(sin2x)^2+2sin2x=-2(sin2x-1/2)^2+1/2,sin2x∈[-1,1].
设t=sin2x,g(t)=-2(t-1/2)^2+1/2,当t∈[-1,1/2]时,g(t)单调递增,当t∈[1/2,1]时,g(t)单调递减.
令sin2x∈[-1,1/2],则2x∈[2kπ-7π/6,2kπ+π/6],即x∈[kπ-7π/12,kπ+π/12],
令sin2x∈[1/2,1],则2x∈[2kπ+π/6,2kπ+5π/6],即x∈[kπ+π/12,kπ+5π/12],
sin2x的单调递增区间为[kπ-π/4,kπ+π/4],
sin2x的单调递减区间为[kπ+π/4,kπ+3π/4],
[kπ-7π/12,kπ-3π/12]包含于[kπ+π/4,kπ+3π/4],[kπ-3π/12,kπ+π/12]包含于[kπ-π/4,kπ+π/4],
[kπ+π/12,kπ+3π/12]包含于[kπ-π/4,kπ+π/4],[kπ+3π/12,kπ+5π/12]包含于[kπ+π/4,kπ+3π/4],
所以f(x)在[kπ-7π/12,kπ-3π/12]上单调递减,f(x)在[kπ-3π/12,kπ+π/12]上单调递增,
f(x)在[kπ+π/12,kπ+3π/12]上单调递减,f(x)在[kπ+3π/12,kπ+5π/12]上单调递增.