已知向量a=(cos3/2x,sin3/2x),b=(cos1/2x,-sin1/2x),且x∈[0,派/2],若f(x)=a·b-2K|a+b|的最小值是-3/2,求K的值.谢咯..
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![已知向量a=(cos3/2x,sin3/2x),b=(cos1/2x,-sin1/2x),且x∈[0,派/2],若f(x)=a·b-2K|a+b|的最小值是-3/2,求K的值.谢咯..](/uploads/image/z/2477144-56-4.jpg?t=%E5%B7%B2%E7%9F%A5%E5%90%91%E9%87%8Fa%3D%28cos3%2F2x%2Csin3%2F2x%29%2Cb%3D%28cos1%2F2x%2C-sin1%2F2x%29%2C%E4%B8%94x%E2%88%88%5B0%2C%E6%B4%BE%2F2%5D%2C%E8%8B%A5f%28x%29%3Da%C2%B7b-2K%7Ca%2Bb%7C%E7%9A%84%E6%9C%80%E5%B0%8F%E5%80%BC%E6%98%AF-3%2F2%2C%E6%B1%82K%E7%9A%84%E5%80%BC.%E8%B0%A2%E5%92%AF..)
已知向量a=(cos3/2x,sin3/2x),b=(cos1/2x,-sin1/2x),且x∈[0,派/2],若f(x)=a·b-2K|a+b|的最小值是-3/2,求K的值.谢咯..
已知向量a=(cos3/2x,sin3/2x),b=(cos1/2x,-sin1/2x),且x∈[0,派/2],若f(x)=a·b-2K|a+b|的最小值是-3/2,求K的值.谢咯..
已知向量a=(cos3/2x,sin3/2x),b=(cos1/2x,-sin1/2x),且x∈[0,派/2],若f(x)=a·b-2K|a+b|的最小值是-3/2,求K的值.谢咯..
楼上解法正确,过程有点错误:已知向量a=(cos3/2x,sin3/2x),b=(cos1/2x,-sin1/2x),且x∈[0,派/2],若f(x)=ab-2K|a+b|的最小值是-3/2,求K的值.a·b =[cos(3/2)x·cos(1/2)x + sin(3/2)x·(-sin1/2x) =[cos(3/2)x·cos(1/2)x - sin(3/2)x·sin(1/2)x =cos[(3/2)x+(1/2)x] = cos(2x) |a+b|^=[cos(3/2)x+cos(1/2)x]^+[sin(3/2)x-sin(1/2)x]^ =2 + 2cos(3/2)xcos(1/2)x - 2sin(3/2)x·sin1/2x) =2+2cos(2x)=4cos^x f(x) =cos(2x)-2K·|2cosx| =2|cosx|^-4K|cosx|-1 =2[|cosx|-K]^-(2K^+1) 如果K≥1,则|cosx|=1时,f(x)有最小值=1-4K=-3/2--->K=5/8,矛盾; 如果K≤0,则|cosx|=0时,f(x)有最小值=-1≠-3/2--->矛盾; 如果0K=±1/2 综上,K=±1/2