f(x)=2sin(2x+π/6) x属于[π/12，π/2] 求值域

Question

为什么sinx在[π/3,π/2]是增函数，[π/2,7π/6]是减函数？

Answer 1

解x属于[π/12，π/2]
即π/12≤x≤π/2
即π/6≤2x≤π
即π/6+π/6≤2x+π/6≤π+π/6
即π/3≤2x+π/6≤7π/6
即-1/2≤sn（2x+π/6）≤1
即-1/2≤y≤1
即值域｛y/-1/2≤y≤1｝

Answer 2

f(x)=sinx的递增、递减区间分别为
[2kπ-π/2，2kπ+π/2]，k∈Z
[2kπ+π/2，2kπ+3π/2]，k∈Z
又2＞0
故当f(x)=2sin(2x+π/6)递增时，
x∈{x|2kπ-π/2≤2x+π/6≤2kπ+π/2，k∈Z}
=[kπ-π/3，2kπ+π/6]，k∈Z
当f(x)=2sin(2x+π/6)递减时，
x∈{x|2kπ+π/2≤2x+π/6≤2kπ+3π/2，k∈Z}
=[kπ+π/6，kπ+2π/3]，k∈Z
而[π/3,π/2]⊆ [2kπ-π/2，2kπ+π/2]，k∈Z
[π/2,7π/6]⊆[2kπ+π/2，2kπ+3π/2]，k∈Z
[π/12，π/2]⊆[kπ+π/6，kπ+2π/3]，k∈Z
所以
f(x)=sinx在[π/3,π/2]上是增函数、[π/2,7π/6]上是减函数；
f(x)=2sin(2x+π/6) 在x∈[π/12，π/2] 上递减，其值域为y∈[ f(π/2)， f(π/12)]=[-1，2√3]
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