1.向量(OA*OB)=(-2sinx*cosx+cos2x)=-sin2x+cos2x
=-√2*sin(2x-π/4).
f(x)=-√2*sin(2x-π/4).
而,x∈[0,π/2].
0≤x≤π/2,
-π/4≤2x-π/4≤3π/4,
要使f(x)有最大值,则,sin(2x-π/4)=-√2/2.
即,f(x)最大=-√2*(-√2/2)=1.
要使f(x)最小值,则,sin(2x-π/4)=1.
即,f(x)最小=-√2*1=-√2.
2.向量OA⊥向量OB,则有
向量(OA*OB)=0,
即,-√2*sin(2x-π/4)=0,
2X-π/4=0,
X=π/8.
而,向量AB=向量(OB-OA)=(-cosx-sinx,1-cos2x),
|AB|=√[(-cosx-2sinx)^2+(1+cos2x)^2]
=√[1+3sin^2x+4sinx*cosx+(2cos^2x)^2]
=√[1+3/2*(1-cos2x)+4sin2x+(cos2x+1)^2].
而,X=π/8.代入上式得,
|AB|=√[1+3/2*(1-cos2x)+4sin2x+(cos2x+1)^2]
=√[1+3/2*(1-√2/2)+4*√2/2+(√2/2+1)^2]
=[√(16+9√2)]/2.