数学已知函数f(x)=sin(x+θ)+cos(x+θ)的定义域为R.(1)当θ=π/2时,求f(x)的单调递增区间.RT

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数学已知函数f(x)=sin(x+θ)+cos(x+θ)的定义域为R.(1)当θ=π/2时,求f(x)的单调递增区间.RT

数学已知函数f(x)=sin(x+θ)+cos(x+θ)的定义域为R.(1)当θ=π/2时,求f(x)的单调递增区间.RT
数学已知函数f(x)=sin(x+θ)+cos(x+θ)的定义域为R.(1)当θ=π/2时,求f(x)的单调递增区间.
RT

数学已知函数f(x)=sin(x+θ)+cos(x+θ)的定义域为R.(1)当θ=π/2时,求f(x)的单调递增区间.RT
f(x)=sin(x+θ)+cos(x+θ)
= √2[√2/2sin(x+θ)+√2/2cos(x+θ)]
=√2[sin(x+θ)cosπ/4+cos(x+θ)sinπ/4]
=√2sin(x+θ+π/4)
当 θ=π/2时,f(x) =√2sin(x+θ+π/4)
=√2sin(x+3π/4)
当 x+3π/4 ∈[2kπ-π/2,2kπ+π/2]时,f(x)单调递增;
此时 x ∈[2kπ-5π/4,2kπ-π/4] 其中 k ∈Z.

你学了导数吗?先对f(x)求导,得新的函数f'(x)。令f'(x)>0解出X,所的即为f(x)的单调增区间。我没记错的话,对f(x)求导得f'(x)应该是f'(x)=cos(x+θ)-sin(x+θ)

f(x)=sin(x+π/2)+cos(x+π/2)
=cos(x)-sin(x)
=√2cos(x+π/4)
x ∈[2kπ-5π/4,2kπ-π/4] 其中 k ∈Z。