The mean value theorem of Flett and divided differences (Q1876697)

Flett's mean value theorem reads as follows: If \(f\) is differentiable on \([a,b]\) and \(f'(a)=f'(b)\), then there exists a point \(c\in(a,b)\) such that \[ f(c)-f(a)=f'(c)(c-a).\tag{11} \] After a careful analysis of divided differences on multiple knots, the authors rewrite as \([a,c,c;f]\) \(=0\) and give condensed representations of other Flett type theorems in terms of divided differences. Several new mean value theorems are obtained. One of them is the following: If \(f\) is continuous on \([a,b]\) and possesses derivatives of order \(n\) at \(a\) and \(b\) such that \(f^{(n)}(a)=f^{(n)}(b)\), then there exists \(c\in(a,b)\) such that in any neighborhood of \(c\) there exist equidistant points \(c_0<\dots<c_n,\;c_0<c<c_n\), with \([a,c_0,\dots,c_n;f]=0\).