Question

y=sin6次方x+cos6次方x的最小正周期,并求函数的最大值和最小值

Answer 1

y=sin^6x+cos^6
=(sin^2x+cos^2x)(sin^4x-sin^2xcos^2x+cos^4x)
=sin^4x-sin^2xcos^2x+cos^4x
=sin^4x+2sin^2xcos^2x+cos^4x-3sin^2cos^2
=(sin^2+cos^2)^2-3/4(2sinxcosx)^2
=-3/4*sin^2(2x)+1
=-3/8*2sin^2(2x)+1
=-3/8 *(1-cos4x)+1
=3/8*cos4x-3/8+1
=3/8*cos4x+5/8
所以最小正周期=2π/4=π/2
最大值=3/8+5/8=1
最小值=-3/8+5/8=1/4