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 6 of 6: Combine both bounds to conclude
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f(x)=1/xf(x)=1/x
Detailed analysis

Together with xf(x)≤1xf(x)\le1 from the third step, we get 1/x≤f(x)≤1/x1/x\le f(x)\le1/x, so f(x)=1/xf(x)=1/x for every x∈R+x\in\mathbb{R}^+, which is the unique solution.