已知向量a=(cos3/2x,sin3/2x),b=(cosx/2,-sinx/2),且x∈[-π/3,π/4]①求a·b 及│a+b│ ②若f(x)=a·b -│a+b│,求f(x)的最大值和最小值
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![已知向量a=(cos3/2x,sin3/2x),b=(cosx/2,-sinx/2),且x∈[-π/3,π/4]①求a·b 及│a+b│ ②若f(x)=a·b -│a+b│,求f(x)的最大值和最小值](/uploads/image/z/2524522-58-2.jpg?t=%E5%B7%B2%E7%9F%A5%E5%90%91%E9%87%8Fa%3D%28cos3%2F2x%2Csin3%2F2x%29%2Cb%3D%28cosx%2F2%2C-sinx%2F2%29%2C%E4%B8%94x%E2%88%88%5B-%CF%80%2F3%2C%CF%80%2F4%5D%E2%91%A0%E6%B1%82a%C2%B7b+%E5%8F%8A%E2%94%82a%2Bb%E2%94%82+%E2%91%A1%E8%8B%A5f%28x%29%3Da%C2%B7b+-%E2%94%82a%2Bb%E2%94%82%2C%E6%B1%82f%28x%29%E7%9A%84%E6%9C%80%E5%A4%A7%E5%80%BC%E5%92%8C%E6%9C%80%E5%B0%8F%E5%80%BC)
已知向量a=(cos3/2x,sin3/2x),b=(cosx/2,-sinx/2),且x∈[-π/3,π/4]①求a·b 及│a+b│ ②若f(x)=a·b -│a+b│,求f(x)的最大值和最小值
已知向量a=(cos3/2x,sin3/2x),b=(cosx/2,-sinx/2),且x∈[-π/3,π/4]
①求a·b 及│a+b│
②若f(x)=a·b -│a+b│,求f(x)的最大值和最小值
已知向量a=(cos3/2x,sin3/2x),b=(cosx/2,-sinx/2),且x∈[-π/3,π/4]①求a·b 及│a+b│ ②若f(x)=a·b -│a+b│,求f(x)的最大值和最小值
①a·b=cos(3/2x)*cos(x/2)-sin(3/2x)*sin(x/2)=cos(3/2x+x/2);
a+b=(cos3/2x+cosx/2,sin3/2x-sinx/2);使用和差化积公式得
a+b=(2cos(3/4x+x/4)*cos(3/4x-x/4),2cos(3/4x+x/4)*sin(3/4x-x/4))
|a+b|=2|cos(3/4x+x/4)| (cos²θ+sin²θ=1)
②f(x)=a·b-|a+b|=cos(3/2x+x/2)-2|cos(3/4x+x/4)|
设3/4x+x/4=t,则f(x)=cos2t-2|cost|=2cos²t-1-2|cost|=2|cost|²-2|cost|-1
其中t=3/4x+x/4是对勾函数,x>0时t≥2√[(3/4x)*(x/4)]=√3/2,则值域为t∈(-∞,-√3/2]∪[√3/2,+∞)
显然t的值域包含完整周期,使cost可以取到[-1,1]上的任意值,则|cost|∈[0,1]
则f(x)=2|cost|²-2|cost|-1=2(|cost|-1/2)²-3/2,
|cost|=0或1时有最大值-1;|cost|=1/2时有最小值3/2
向量a.向量b=cos3x/2*cosx/2-sin3x/2*sinx/2.
(1) a.b=cos(3x+x)/2=cos2x. ∵x∈[-π/3,π/4], 2x∈[-2π/3,π/2]
a.b=cos2x=cos(-2π/3=cos2π/3=cos(π-π/3)=-cosπ/3=-1/2.
或,a.b=cos2x=cosπ/2=0.
a+b=...
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向量a.向量b=cos3x/2*cosx/2-sin3x/2*sinx/2.
(1) a.b=cos(3x+x)/2=cos2x. ∵x∈[-π/3,π/4], 2x∈[-2π/3,π/2]
a.b=cos2x=cos(-2π/3=cos2π/3=cos(π-π/3)=-cosπ/3=-1/2.
或,a.b=cos2x=cosπ/2=0.
a+b=(cos3x/2+cosx/2,sin3x/2-sinx/2).
=2(cos2xcosx,cos2xsinx).
|a+b|=√2^2[cos^2(2x)cos^2x+cos^2(2x)sin^2x]
=2√[cos^2(2x)(cos^2x+sin^2x)]
=2cos2x.
(2) f(x)=a.b-|a+b|=cos2x-2cos2x=co2x(1-2)=-cos2x
前面解出,cos2x=-1/2,∴f(x)max=-(-1/2)=1/2;
cos2x=0, f(x)min=0.
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