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.



A generalized mean-value theorem
Let $q(x)$ be the quadratic through the $(x\_i, f(x\_i))$. Set $g(x) = f(x) - q(x)$. Then $g(x\_1) = g(x\_2) = g(x\_3) = 0$ and you want to show $g''(x)=0$ for some $x$. Apply Rolle's theorem to get that $g'$ has zeroes $y\_1$ and $y\_2$ in $(x\_1, x\_2)$ and $(x\_2, x\_3)$ respectively, then apply it again to get that $g''$ has a zero in $(y\_1, y\_2)$.




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I can sort of give you a pointer to the literature. Rename your points $x\_1$, $x\_2$, ..., $x\_{n}$ and $z$ with $x\_1 < x\_2 < \ldots < x\_{n}$. Let $q(t)$ be the degree $n-1$ Lagrange interpolating polynomial for $f$ through $(x\_i, f(x\_i))$. Then the degree $n$ Lagrange interpolating polynomial through the $(x\_i, f(x\_i))$ and $(z, f(z))$ is 
$$r(t) = q(t) + \frac{f(z)-q(z)}{\prod (z-x\_i)} \prod (t-x\_i).$$
You want to know that there is some $c \in (\min(x\_1,z), \max(x\_n,z))$ where $f^{(n)}(c) = r^{(n)}(c)$. 
In other words, where $f^{(n)}(c) = \frac{f(z) - q(z)}{\prod (z-x\_i)} n!$. You can rearrange this to give
$$f(z) = q(z) + \frac{f^{(n)}(c)}{n!} \prod (z-x\_i).$$
This is a known formula for the error in Lagrange interpolation. Here is a [Wikipedia article](http://en.wikipedia.org/wiki/Polynomial_interpolation#Interpolation_error). The case where $x\_1=x\_2=\cdots = x\_n$ is Taylor's theorem with the Lagrange form of the remainder; Tim Gowers has a [nice blog post](https://gowers.wordpress.com/2014/02/11/taylors-theorem-with-the-lagrange-form-of-the-remainder/). Comments on Prof. Gowers' post suggest that Atkinson’s Intro to Numerical Analysis would give you a more official source; I haven't read it.


I don't know a source where the $n+1$ points are treating on equal footing, but my ignorance of analysis literature is no sign that it doesn't exist.


