已知向量a=[cos(3x/2),sin(3x/2)],已知向量b=[cos(x/2),-sin(x/2)],x属于[0
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已知向量a=[cos(3x/2),sin(3x/2)],已知向量b=[cos(x/2),-sin(x/2)],x属于[0,兀/3]
1)求F(x)=向量a*向量b/|向量a+向量b|的最大值
2)若不等式 入*向量a*向量b-1/2|向量a+向量b|+入-1小于等于0对x属于[0.,兀/3]恒成立,求实数入的取值范围
1)求F(x)=向量a*向量b/|向量a+向量b|的最大值
2)若不等式 入*向量a*向量b-1/2|向量a+向量b|+入-1小于等于0对x属于[0.,兀/3]恒成立,求实数入的取值范围
(1)向量a=(cos3x/2,sin3x/2),
向量b=(cosx/2),-sinx/2),
向量a·b=(cos3x/2)*(cosx/2)+(sin3x/2)*(-sinx/2),
=cos(3x/2+x/2)=cos2x,
向量a+b=(cos3x/2+cosx/2,sin3x/2-sinx/2),
|a+b|=√[(cos3x/2+cosx/2)^2+(sin3x/2-sinx/2)^2]
=√{(2cos2xcosx/2)*2+(2cos2xsinx/2)]
=2√(cos2x)^2*[cosx/2)^2+(sinx/2)^2]
=2|cos2x|,
F(x)=a·b/|a+b|
=cos2x/(2|cos2x|),
x∈[0,π/3],
2x∈[0,2π/3],
当2x∈[0,π/2]时,cos2x>0,
F(x)=1/2,最大.
(2)λa·b-(1/2)|a+b|+λ-1≤0,x∈[0,π/3],
λcos2x+λ≤cos2x/(4|cos2x|)+1,
λ(1+cos2x)≤cos2x/(4|cos2x|)+1,
2x∈[0,2π/3],
1+cos2x≥0,
λ≤[cos2x/(4|cos2x|)+1]/(1+cos2x),
当2x∈[0,π/2]时,
λ≤(5/4)(1+cos2x),
1+cos2最大为2,
λ≤5/8,
2x∈[π/2,2π/3]时,
λ≤(3/4)/(1+cos2x),
(1+cos2x)最大为1/2,(1-1/2)
λ≤3/2,
取其交集,
λ≤5/8.
向量b=(cosx/2),-sinx/2),
向量a·b=(cos3x/2)*(cosx/2)+(sin3x/2)*(-sinx/2),
=cos(3x/2+x/2)=cos2x,
向量a+b=(cos3x/2+cosx/2,sin3x/2-sinx/2),
|a+b|=√[(cos3x/2+cosx/2)^2+(sin3x/2-sinx/2)^2]
=√{(2cos2xcosx/2)*2+(2cos2xsinx/2)]
=2√(cos2x)^2*[cosx/2)^2+(sinx/2)^2]
=2|cos2x|,
F(x)=a·b/|a+b|
=cos2x/(2|cos2x|),
x∈[0,π/3],
2x∈[0,2π/3],
当2x∈[0,π/2]时,cos2x>0,
F(x)=1/2,最大.
(2)λa·b-(1/2)|a+b|+λ-1≤0,x∈[0,π/3],
λcos2x+λ≤cos2x/(4|cos2x|)+1,
λ(1+cos2x)≤cos2x/(4|cos2x|)+1,
2x∈[0,2π/3],
1+cos2x≥0,
λ≤[cos2x/(4|cos2x|)+1]/(1+cos2x),
当2x∈[0,π/2]时,
λ≤(5/4)(1+cos2x),
1+cos2最大为2,
λ≤5/8,
2x∈[π/2,2π/3]时,
λ≤(3/4)/(1+cos2x),
(1+cos2x)最大为1/2,(1-1/2)
λ≤3/2,
取其交集,
λ≤5/8.
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