函數(shù)f(x)=sin(x/2)cos(x/2)+cos²(x/2)
=(sin(x/2)+cos(x/2))cos(x/2)
=√2(√2/2sin (x/2)+√2/2cos(x/2))cos(x/2)
=√2(sin (x/2) cosπ/4+cos (x/2) sinπ/4)cos(x/2)
=√2sin(x/2+π/4)cos(x/2)
=√2/2sin(x+π/4)+ 1/2
1、
f(x) = √2/2sin(x+π/4)+ 1/2
sin(x),cos(x)的定義域?yàn)镽,值域?yàn)椤?1,1〕
即：-π/2+2kπ≤x+π/4≤π/2+2kπ
函數(shù)f(x)的單增區(qū)間為-3π/4+2kπ≤x≤π/4+2kπ.(k∈z).
即：【-3π/4+2kπ,π/4+2kπ】
2、
-π/4≤x≤π
0≤x+π/4≤5π/4
所以x+π/4=5π/4,sin(x+π/4)最小=-√2/2
函數(shù)f(x)的最小值為
f(x) =√2/2sin(x+π/4)+ 1/2=-√2/2×√2/2+ 1/2=-1/2+1/2=0
x+π/4=π/2,sin(x+π/4)最大=1
函數(shù)f(x)的最大值為
f(x) =√2/2sin(x+π/4)+ 1/2=√2/2+ 1/2