Let R+ denote the set of positive real numbers. Find all functions f:R⟶R such that for each x∈R+, there is exactly one y∈R+ satisfying:
xf(y)+yf(x)≤2.
译文:
设R+表示所有正实数构成的集合.求所有函数f:R+→R+,使得对任意x∈R+,恰好有一个y∈R+满足条件:xf(y)+yf(x)≤2.
Let R+ denote the set of positive real numbers. Find all functions f:R⟶R such that for each x∈R+, there is exactly one y∈R+ satisfying:
xf(y)+yf(x)≤2.
译文:
设R+表示所有正实数构成的集合.求所有函数f:R+→R+,使得对任意x∈R+,恰好有一个y∈R+满足条件:xf(y)+yf(x)≤2.
对任意正实数x,函数f(x)=1/x满足题意.代入条件使得xf(y)+yf(x)=x/y+y/x≥2=2,但xf(y)+yf(x)≤2,所以,对任意x∈R+,存在唯一实数y=x满足xf(y)+yf(x...
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