f（x）=sin^6X+cos^6X

来源：百度知道 编辑：UC知道 时间：2024/06/06 00:04:46
1)判断函数的奇偶性
（2）求f（x）在【-π/16,π/24]上的最大值和最小值，并求取得最大最小值时x的值
（3）求f（x）的周期和单调区间

f(x)=(sin^2x+cos^2x)(sin^4x-sin^2x*cos^2x+cos^4x)
=sin^4x-sin^2x*cos^2x+cos^4x
=(sin^2x+cos^2x)^2-3sin^2x*cos^2x
=1-3/4*(4sinx*cosx)^2
=1-(3/4)sin^2 2x
=1-(3/4)(1-cos4x)/2
=-5/8+(3/8)cos4x
-π/16<=x<=π/24
-π/4<=4x<=π/6
所以x=0时cos4x有最大值，f(x)有最大值
4x=-π/4,x=-π/16时cos4x有最小值，f(x)有最小值

T=2π/4=π/2

f(x)=-5/8+(3/8)cos4x
当2kπ<4x<2kπ+π时,kπ/2<x<2kπ+π/2，f(x)递减
2kπ-π<4x<2kπ时，kπ/2-π/2<x<kπ/2,f(x)递增
-π/16<=x<=π/24
所以0<x<π/24，f(x)递减
-π/16<x<0,f(x)递增