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An extension of a theorem of Flett. (English) Zbl 0955.26003
T. M. Flett [Math. Gaz. 42, 38-39 (1958)] proved the following variant of the Lagrange mean value theorem: If \(f\) is differentiable in \([a,b]\) with \(f'(a)=f'(b)\), then there is an \(\eta \in (a,b)\) such that \(f(\eta)-f(a) = (\eta-a) f'(\eta)\). This was refined by P. K. Sahoo and T. Riedel [“Mean value theorems and functional equations”, World Scientific Publishing Co., Inc., River Edje, NJ (1998)] as follows: If \(f\) is differentiable in \([a,b]\), then there is an \(\eta\in(a,b)\) such that \(f(\eta)-f(a)=(\eta-a)f'(\eta)-\frac 1 2 \frac{f'(b)-f'(a)}{b-a} (\eta-a)^2\). The question arises if there is an extension of this mean value theorem for functions of higher differentiability, in analogy to Taylor’s theorem. This question is answered in the present paper. The author proves, quite elegantly, the following: If \(f\) is \(n\) times differentiable in \([a,b]\), then there exists an \(\eta\in (a,b)\) such that \[ f(\eta)-f(a) = \sum_{k=1}^n (-1)^{k-1} \frac{1}{k!} f^{(k)}(\eta) (\eta-a)^k + (-1)^n \frac{1}{(n+1)!} \frac{f^{(n)}(b)-f^{(n)}(a)}{b-a} (\eta-a)^{n+1}. \]

26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
26A06 One-variable calculus
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