f(x)=2sin(sinx+cosx) 求值域

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f(x)=2sin(sinx+cosx)求值域f(x)=2sin(sinx+cosx)求值域f(x)=2sin(sinx+cosx)求值域)f(x)=2sinxcosx+2sin²x=sin

f(x)=2sin(sinx+cosx) 求值域
f(x)=2sin(sinx+cosx) 求值域

f(x)=2sin(sinx+cosx) 求值域
)f(x)=2sinxcosx+2sin²x
=sin2x+(1-cos2x)
=sin2x-cos2x+1
=√2sin(x-π/4)+1
-√2

sinx+cosx=√2sin(x+π/4)∈[-√2,√2]
f(x)=2sin(sinx+cosx)∈[-2sin√2,2sin√2]

f(x)=2sin[2^1/2sin(x+pai/4)]=2sint
t=2^1/2sin(x+pai/4)
t:[-2^1/2,2^1/2]真包含于[-pai/2,pai/2]
x=-2^1/2,fmin=-2*2^1/2
x=2^1/2,fmax=2*2^1/2
f(x)=[-2*2^1/2,2*2^1/2]