已知函数f(x)=2sin(x/4)cos(x/4)-2√3sin²(x/4)+√3,且g(x)=f(x+π/3)(1)判断g(x)的奇偶性 (2)求g(x)的单调递增区间
来源:学生作业帮助网 编辑:作业帮 时间:2024/11/08 17:41:58
已知函数f(x)=2sin(x/4)cos(x/4)-2√3sin²(x/4)+√3,且g(x)=f(x+π/3)(1)判断g(x)的奇偶性 (2)求g(x)的单调递增区间
已知函数f(x)=2sin(x/4)cos(x/4)-2√3sin²(x/4)+√3,且g(x)=f(x+π/3)
(1)判断g(x)的奇偶性 (2)求g(x)的单调递增区间
已知函数f(x)=2sin(x/4)cos(x/4)-2√3sin²(x/4)+√3,且g(x)=f(x+π/3)(1)判断g(x)的奇偶性 (2)求g(x)的单调递增区间
f(x)=sin(x/2) -√3[1-cos(x/2)]+√3
=2[(1/2)sin(x/2) +(√3/2)cos(x/2)]
=2sin(x/2+π/3)
(1) g(x)=f(x+π/3)=2sin[(x+π/3)/2 +π/3)]=2sin(x/2 +π/2)=2cos(x/2)
所以 g(x)是偶函数.
(2)令 -π+2kπ≤x/2≤2kπ,得
-2π+4kπ≤x≤4kπ,
所以 增区间为
[-2π+4kπ,4kπ],k是整数.
1
f(x)=2sin(x/4)cos(x/4)-2√3sin²(x/4)+√3
f(x)=2sin(x/4)cos(x/4)-√3(-2sin²(x/4)+1)
=sin(x/2)-√3cos(x/2)
=2sin(x/2-π/3)
g(x)=2sin[(x+π/3)/2-π/3]
=2sin(x/2-π/6)
对称轴x...
全部展开
1
f(x)=2sin(x/4)cos(x/4)-2√3sin²(x/4)+√3
f(x)=2sin(x/4)cos(x/4)-√3(-2sin²(x/4)+1)
=sin(x/2)-√3cos(x/2)
=2sin(x/2-π/3)
g(x)=2sin[(x+π/3)/2-π/3]
=2sin(x/2-π/6)
对称轴x/2-π/6=kπ+π/2
x=2kπ+4π/3≠0
∴g(x)是非奇非偶函数
2
2kπ-π/2≤x/2-π/6≤2kπ+π/2
4kπ-2π/3≤x≤4kπ+4π/3
收起