Question

设f(x)=sin∧4x+cos∧4x,求其单调区间

Answer 1

原题是:设f(x)=(sinx)^4+(cosx)^4,求其单调区间.
f(x)=(sinx)^2+(cosx)^2)^2-2(sinx)^2(cosx)^2
=1^2-(1/2)(2sinxcosx)^2
=1-(1/2)(sin2x)^2
=1-(1/4)(1-cos4x)
=(1/4)cos4x+(3/4)
单增:2kπ-π≤4x≤2kπ,k∈Z
kπ/2-(π/4)≤x≤kπ/2,k∈Z
单减:2kπ≤4x≤2kπ+π,k∈Z
kπ/2≤x≤kπ/2+(π/4),k∈Z
所以f(x)的单调递增区间是:[kπ/2-(π/4),kπ/2],k∈Z
单调递减区间是:[kπ/2,kπ/2+(π/4)],k∈Z

希望能帮到你！