The following problem has been proposed (#1837, October 2010) by Duong Viet Thong, Nam Dinh University of Technology Education, Nam Dinh City, Vietnam.

Problem

Let f: [0,1]\rightarrow R be a continuous function such that \int_{1}^{2}f(x)dx=0. Prove there exists c in the open interval (1,2) such that cf(c)=\int_{c}^{2}f(x)dx.

Solution

The solution is Ángel Plaza and Sergio Falcón, Department of Mathematics, Universidad de Las Palmas de Gran Canaria, Las Palmas, Spain.

Consider function F(t) = t \int_{t}^{2}f(x)d(x) . F(1)= F(2)=0 and also F is continuous on [1,2] and differentiable on (1,2). Rolle's Theorem applies and implies the existence of c\in(0,1) such that F'(c)=0.

But F'(t) = \int_{t}^{2}f(x)d(x) - tf(t). So that for c from Rolle's Theorem, \int_{c}^{2}f(x)d(x) - cf(c) = 0.