求函数y=cos^2x-cosx+2的递增及递减区间

Question

求函数y=cos^2 x-cos x+2的递增及递减区间

Answer 1

y=cos^2 x-cos x+2
=(cosx-1/2)²+7/4
设t=cosx∈[-1,1]
y=(t-1/2)²+7/4
当t∈[-1,-1/2]时,y=(t-1/2)²+7/4递减
此时x∈[2π/3+2kπ,π+2kπ],k∈Z,t=cosx递减
∴[2π/3+2kπ,π+2kπ],为原函数的递增区间
x∈[π+2kπ,4π/3+2kπ],k∈Z,t=cosx递增
∴[π+2kπ,4π/3+2kπ],为原函数的递减区间
当t∈[-1/2,1]时,y=(t-1/2)²+7/4递增
此时x∈[4π/3+2kπ,2π+2kπ],k∈Z,t=cosx递增
∴[4π/3+2kπ,2π+2kπ],k∈Z 为原函数的递增区间
x∈[2kπ,2π/3+2kπ],k∈Z,t=cosx递减
∴[2kπ,2π/3+2kπ],k∈Z为原函数的递减区间
∴原函数的递减区间是[2kπ,2π/3+2kπ],[π+2kπ,4π/3+2kπ],k∈Z
原函数的递增区间是[4π/3+2kπ,2π+2kπ],[2π/3+2kπ,π+2kπ],