若非零函数f(x)对任意实数a,b均有f(a+b)=f(a)f(b),且当x<0时,f(x)>1

当f(4)=1/16时,解不等式f(x-3)×f(5-x2)&lt;=1/4
2025-04-30 21:06:30
推荐回答(2个)
回答1:

由f(a+b)=f(a)f(b)得:f(x-3)=f(x)f(-3)

f(5-x2)=f(-2x)f(5) ∴f(x-3)×f(5-x2)==f(x)f(-3)f(-2x)f(5﹚
=f(-x)f﹙2﹚
1/4=f(4)4 f(4)=f(2+2)=f(2)*f(2) f(4)=1/16∴f(2)≡1/4或-1/4

①当f(2)≡1/4,f(-x)≦4*f(2)

∴f(-x)≦1,无解

②当f(2)≡-1/4,f(-x)≧1-----------------x>0

综上所述,x>0

回答2:

设a=b=2 f(2+2)=f(4)=f(2)f(2)=1/16 f (2)=+-1/4 有条件知f(2)=1/4 f(x-3)×f(5-x2)=f(x-3+5-x2)=1/4

x-3+5-x2>=0 -1<=x<=2