已知向量a=(cos3x/2,sin3x/2),b=(cosx/2,-sinx/2),且x∈[-π/3,π/4].

题目

已知向量a=(cos3x/2,sin3x/2),b=(cosx/2,-sinx/2),且x∈[-π/3,π/4].
1）求a*b及|a+b|的表达式
2）若f（x）=a*b-|a+b|.求发（x）最大值和最小值

答案

ab=cos(3x/2)*cos(x/2)-sin(3x/2)*sin(x/2)
=cos[(3x+x)/2]
=cos(2x).
a+b=(cos(3x/2)+cos(x/2),sin(3x/2)-sin(x/2)),
|a+b|=√[(cos(3x/2)+cos(x/2))^2+(sin(3x/2)-sin(x/2))^2]
=√[2(1+cos2x)]
=2*|cosx|,
因为,x∈[-π/3,π/4].则有,cosx>0,
即,
|a+b|=2*|cosx|=2cosx.
2.若f（x）=a*b-|a+b|.则有,
f(x)=cos2x-2cosx,
=2cos^2x-1-2cosx
=2(cosx-1/2)^2-3/2.
而,x∈[-π/3,π/4].则有,
1)当X=0时,cos0=1,则f(x)=2(1-1/2)^2-3/2=-1.
2)当X=π/4时,cosπ/4=√2/2,则f(x)=2*(√2/2-1/2)^2-3/2=-√2.
则,f(x)最大值=-1,f(x)最小值=-√2.