已知函数f (x)对任意x,y∈R,总有

题目

已知函数f (x)对任意x,y∈R,总有
f (x)＋f ( y)＝f (x＋y),且当x＞0时,
f (x)＜0,f (1)＝2÷3
(1)求证f (x)是R上的减函数；
(2)求f (x)在[－3,3]上的最大值和最小值.

答案

(1)取x=0,y=0,则f(0)+f(0)=f(0+0)f(0)=0;取y为-x,则f (x)＋f ( -x)＝f (x-x)=f(0)=0,得f(x)为奇函数;任取x1>x2f (x1)＋f ( -x2)＝f (x1-x2),f (x1)-f ( x2)＝f (x1-x2)0) ( 当x＞0时,f (x)＜0)所以f(x)是R上的减函数...