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Expression attains positive and negative values

Source: St. Petersburg 2023 10.5

August 12, 2023
algebra

Problem Statement

Let x0,x1,…,xn−1,xn=x0x_0, x_1, \ldots, x_{n-1}, x_n=x_0 be reals and let f:R→Rf: \mathbb{R} \rightarrow \mathbb{R} be a function. The numbers yiy_i for i=0,1,…,n−1i=0,1, \ldots, n-1 are chosen such that yiy_i is between xix_i and xi+1x_{i+1}. Prove that ∑i=0n−1(xi+1−xi)f(yi)\sum_{i=0}^{n-1}(x_{i+1}-x_i)f(y_i) can attain both positive and negative values, by varying the yiy_i.
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