MathLabs

Problem 2

Let R+\mathbb{R}^+ denote the set of positive real numbers. Find all functions f:R+→R+f:\mathbb{R}^+\to\mathbb{R}^+ such that for each x∈R+x\in\mathbb{R}^+, there is exactly one y∈R+y\in\mathbb{R}^+ satisfying xf(y)+yf(x)≤2xf(y)+yf(x)\le2
Step 1 of 6: Check that f(x)=1/xf(x)=1/x works
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f(x)=1/xf(x)=1/x
Detailed analysis

For f(x)=1/xf(x)=1/x, taking y=xy=x gives xf(y)+yf(x)=1+1=2≤2xf(y)+yf(x)=1+1=2\le2, so y=xy=x is a friend of xx; one can check this is in fact the unique such yy, so f(x)=1/xf(x)=1/x satisfies the condition.