α is a real number. Find all continuous real-valued functions f **:**[0, 1] → (0, ∞) such that ∫_{0}^{1} f(x) dx = 1, ∫_{0}^{1} x f(x) dx = α, ∫_{0}^{1} x^{2} f(x) dx = α^{2}.

**Solution**

We have ∫_{0}^{1} (α - x)^{2}f(x) dx = α^{2} - 2α^{2} + α^{2} = 0. But the integrand is positive, except possibly for one point of the range, so the integral must also be positive. Contradiction. So there are no functions with this property.

© John Scholes

jscholes@kalva.demon.co.uk

5 Feb 2002