Question

函数y=sinx(sinx+cosx)值域 x为零到二分之派

Answer 1

y=sinx(sinx+cosx)
=(sinx)^2+sinxcosx
=(1/2)sin2x-(1/2)(cosx)^2+1/2
=(√2/2)sin(2x-π/4)+1/2
若0<=x<=π/2，则-π/4<=2x-π/4<=3π/4，-√2/2<=sin(2x-π/4)<=1，0<=y<=(√2+1)/2
所以，函数y=sinx(sinx+cosx)值域是[0,(√2+1)/2]

Answer 2

=sin²x+sinxcosx=(1/2)(1- cos2x)+(1/2)sin2x
=1/2 + (1/2)[sin2x- cos2x]
=1/2 + √2/2 sin(2x- π/4)
x∈[0,π/2] 2x-π/4∈[-π/4,3π/4]
所以，当：x=0 y=0
x=3π/8 y=(√2+1)/2]
值域为
[0,(√2+1)/2]

Answer 3

y=sinx(sinx+cosx)=sin²x+sinxcosx=根号2sin(2x-π/4)+1/2
当x∈【0，π/2]时，2x-π/4∈【-π/4,3π/4],sin(2x-π/4)∈【-根号2/2,1]
所以y∈【-1/2,根号2+1/2]

Answer 4

y=sinx(sinx+cosx)
=sin²x+sinxcosx
=(1-cos2x)/2+(sin2x)/2
=(√2/2)sin(2x-π/4)+1/2
x∈[0,π/2] 2x-π/4∈[-π/4,3π/4]
所以
值域为
[-1/2,(√2+1)/2]