Zur Übereinstimmung der Mittelwertstellen von Funktionen und ihren Ableitungen. (On the coincidence of the mean-values of functions and their derivatives) (Q1179122)

Let \(f:\mathbb{R}\to\mathbb{R}\) be twice continuously differentiable, and assume that \(f'(x)f''(x)\) is never 0. The theorem proved in this brief note is that for such functions \(f\), the mean-value \(m_ 1\) in the sense of the derivative: \(f'(m_ 1)=(f(b)-f(a))/(b-a)\), coincides with the mean value \(m_ 2\) in the sense of the integral: \(f(m_ 2)=\left(\int^ b_ af(x)dx\right)/(b-a)\), for all intervals \((a,b)\), if and only if for some \(\alpha,\beta\neq0\) and \(\mu\) in \(\mathbb{R}\), \(f(x)=\alpha \exp(\beta x)+\mu\).