定义域在R上的函数f(x)满足f(x)=f(x+2),当x∈[3,5]时,f(x)=2-|x-4|.说明①f(sinπ/6)＜f(cosπ/6)②f(sin1)＞f(cos1)③f(cos2π/3)＜f(xin2π/3)④f(cos2)＞f(sin2)是否正确①②③④都要证明

Question

定义域在R上的函数f(x)满足f(x)=f(x+2),当x∈[3,5]时,f(x)=2-|x-4|.说明①f(sinπ/6)＜f(cosπ/6)
②f(sin1)＞f(cos1) ③f(cos2π/3)＜f(xin2π/3)④f(cos2)＞f(sin2)是否正确
①②③④ 都要证明

Answer 1

x∈[3,4], |x-4|=4-x, f(x)=2-(4-x)=2-4+x=x-2
x∈[4,5], |x-4|=x-4,f(x)=2-(x-4)=2-x+4=6-x
f(x)=f(x+2), 令y=x+2, x=y-2, f(y-2)=f(y)
所以 f(x-2)=f(x)=f(x+2)
x∈[-1,0],[1,2], f(x)=x-2
x∈[0,1],[2,3], (x)=6-x
f(sinπ/6)=f(1/2)=f(1/2+4)=6-(1/2+4)
f(cosπ/6)=f(√3/2)=f(√3/2+4)=6-(√3/2+4)
f(sinπ/6)>f(cosπ/6) 题目错了吗?