TITLE: A generalized mean-value theorem
QUESTION [8 upvotes]: I'm pretty sure that if the function $f$ is continuous on $[x_1,x_3]$ and twice-differentiable on $(x_1,x_3)$, with $x_1 < x_2 < x_3$, then there must exist $x$ in $(x_1,x_3)$ for which $f''(x)$ equals $$\dfrac{((x_2-x_1)(f(x_3)-f(x_1)) - (f(x_2)-f(x_1))(x_3-x_1))}{ ((x_2-x_1)(x_3-x_2)(x_3-x_1))}.$$  (This complicated expression gives the constant second-derivative of the unique polynomial of degree 2 passing through the points $(x_i,f(x_i))$ for $1 \leq i \leq 3$.)
A simple proof would be nice, but even nicer would be a pointer to the literature. Surely this result is known, and indeed is just a special case of a known result for curves passing through $n$ prescribed points.

REPLY [3 votes]: The stated result is a special case of the Schwarz mean value theorem, which plays a crucial role in Dörge's proof of the Hilbert irreducibility theorem. References I know of are the books Diophantine Geometry by Lang (p. 148), Selected Topics on Polynomials by Schinzel (p. 174), and Generic Polynomials by Jensen, Ledet and Yui (p. 69).