作业帮已知向量a=（cos3/2x,sin3/2x）,b=（cosx/2,-sinx/2）,且x属于[0,π/2],若f（x）=a·b-2λ|a+b|的最小值是-3/2,求实数λ的值(化简步骤稍微详细一点,谢谢)
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![已知向量a=（cos3/2x,sin3/2x）,b=（cosx/2,-sinx/2）,且x属于[0,π/2],若f（x）=a·b-2λ|a+b|的最小值是-3/2,求实数λ的值(化简步骤稍微详细一点,谢谢)](/uploads/image/z/2625330-66-0.jpg?t=%E5%B7%B2%E7%9F%A5%E5%90%91%E9%87%8Fa%3D%EF%BC%88cos3%2F2x%2Csin3%2F2x%EF%BC%89%2Cb%3D%EF%BC%88cosx%2F2%2C-sinx%2F2%EF%BC%89%2C%E4%B8%94x%E5%B1%9E%E4%BA%8E%5B0%2C%CF%80%2F2%5D%2C%E8%8B%A5f%EF%BC%88x%EF%BC%89%3Da%C2%B7b-2%CE%BB%7Ca%2Bb%7C%E7%9A%84%E6%9C%80%E5%B0%8F%E5%80%BC%E6%98%AF-3%2F2%2C%E6%B1%82%E5%AE%9E%E6%95%B0%CE%BB%E7%9A%84%E5%80%BC%28%E5%8C%96%E7%AE%80%E6%AD%A5%E9%AA%A4%E7%A8%8D%E5%BE%AE%E8%AF%A6%E7%BB%86%E4%B8%80%E7%82%B9%2C%E8%B0%A2%E8%B0%A2%29)
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已知向量a=（cos3/2x,sin3/2x）,b=（cosx/2,-sinx/2）,且x属于[0,π/2],若f（x）=a·b-2λ|a+b|的最小值是-3/2,求实数λ的值(化简步骤稍微详细一点,谢谢)
已知向量a=（cos3/2x,sin3/2x）,b=（cosx/2,-sinx/2）,且x属于[0,π/2],若f（x）=a·b-2λ|a+b|的最小值是-3/2,求实数λ的值(化简步骤稍微详细一点,谢谢)
已知向量a=（cos3/2x,sin3/2x）,b=（cosx/2,-sinx/2）,且x属于[0,π/2],若f（x）=a·b-2λ|a+b|的最小值是-3/2,求实数λ的值(化简步骤稍微详细一点,谢谢)
a*b=cos[(3/2)x]cos(x/2)-sin[(3/2)x]sin(x/2)=cos2x,
|a+b|^2=[cos(3/2x)+cos(x/2)]^2+[sin(3/2x)-sin(x/2)]^2
=2+2cos2x=4(cosx)^2,
∴f(x)=cos2x-4λ|cosx|
=2(cosx)^2-4λ|cosx|-1
=2[|cosx|-λ]^2-2λ^2-1,
x∈[0,π/2],cosx∈[0,1],
λ∈[0,1]时f(x)|min=-2λ^2-1=-3/2,λ^2=1/4,λ=1/2;
λ1时f(x)|min=1-4λ=-3/2,4λ=5/2,λ=5/8(舍）.
综上,λ=1/2.