Question

若函數(shù)f(x)=2asin(2x+θ)的值域?yàn)椋?2,2］,且在區(qū)間［,］上是單調(diào)遞減函數(shù).

(1)求常數(shù)a的值；

(2)求角θ的值.

Answer 1

解：(1)∵sin(2x+θ)∈［-1,1］,且f(x)∈［-2,2］,知2|a|=2,∴a=±1.

(2)a=1時(shí)，f(x)=2sin(2x+θ),其周期T=π.

∵f(x)在［,］內(nèi)單調(diào)遞減，且-()=為半個(gè)周期，

故必有f(x) _max =f()=2,即2sin(θ-)=2.

∴θ-=2kπ+,θ=2kπ+(k∈Z),a=-1時(shí)，f(x)=-2sin(2x+θ)在［,］上單調(diào)遞減，故f(x)_max=f()=2,則sin(θ-)=-1,θ-=2kπ-,θ=2kπ+(k∈Z).

綜上知θ=kπ+(k∈Z).