Question

急求答案 ！已知函数f(x)=sinx+cosx (1)若f(x)=2f(x+派/2),求tan(x+派/4)的值, (2)求函数g(x)=f(x)f...

急求答案 ！已知函数f(x)=sinx+cosx (1)若f(x)=2f(x+派/2),求tan(x+派/4)的值, (2)求函数g(x)=f(x)f(x+派/2)的最小值及相应的x值的集合

Answer 1

1
f(x)=2f(x+π/2)
sinx+cosx=2(-cosx+sinx)
sinx=3cosx
tanx=1/3
tan(x+45)=（tanx+1）/1-tan^2x=3/2
2/
gx=fxf(x+π/2)
=（sinx+cosx)(-cosx+sinx）
=sin^2x-cos^2x
=-cos2x
min=-1
2x=kπ+π/4时取得

Answer 2

(1):
f(x)=sin(x)+cos(x)) = √2(sinx*√2/2+cosx*√2/2) = √2sin(x+π/4)
所以f(x)=2f(x+π/2) = 2√2sin(x+π/2+π/4) = 2√2cos(π/4+x)
即 √2sin(x+π/4) = 2√2cos(π/4+x)
即 tan(x+π/4) = 2

(2)
g(x) = f(x)f(x+π/2) = √2sin(x+π/4) * √2sin(x+π/4+π/2)
=2sin(x+π/4)cos(x+π/4)
=sin(2x+π/2)=cos(2x)
最小值：-1 相应x的集合{x|x=kπ+π/2，k为整数}
求解完毕