f(x)=2sin^2x+cos^2x-sinxcosx (1)求f(x)的周期,及单调区间(2)求f(x)的最大值,并

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  • (1)

    f(x)=2sin^2x+cos^2x-sinxcosx

    =sin^2x+sin^2x+cos^2x-sinxcosx

    =1+sin^2x-sinxcosx

    =1+1/2-1/2cos2x-1/2sin2x

    =3/2-1/2(sin2x+cos2x)

    =3/2-根号2/2sin(2x+π/4)

    2x的周期为2π,即x的周期为π

    2nπ-π/2≤2x+π/4≤2nπ+π/2时,即nπ-3π/8≤x≤nπ+π/8时,根号2/2sin(2x+π/4)递增,f(x)==3/2-根号2/2sin(2x+π/4)单调减;

    2nπ+π/2≤2x+π/4≤2nπ+3π/2时,即nπ+π/8≤x≤nπ+5π/8时,根号2/2sin(2x+π/4)递减,f(x)==3/2-根号2/2sin(2x+π/4)单调增.

    (2)求f(x)的最大值,并求此时自变量x的集合

    当sin(2x+π/4)=-1时,f(x)有最大值:

    f(x)max=3/2-根号2/2*(-1)=(3+根号2)/2

    此时,2x+π/4=2nπ+3/2π

    即x=nπ+5π/8,(其中n为整数)