I have a question here:
A finite sequence of real numbers $c_1, c_2, \dots, c_{n−1}$ is called saw– like if we have $(−1)^k(c_k − c_{k+1}) \leq 0$ for all $k = 1, \dots , n − 2$ or if we have $(−1)^k(c_k − c_{k+1}) \geq 0$ for all $k = 1, \dots , n − 2$. Prove that $c_1, c_2, \dots , c_{n−1}$ is a saw–like sequence if and only if there exist a polynomial $f (x)$ of degree $n$ with real coefficients such that
(1) $x_1 \leq \dots \leq x_{n−1}$ are critical points of $f$ ;
(2) $f (x_k) = c_k$ for $k = 1, \dots , n − 1$.
I've proven one direction,which is "if we have such a polynomial $f$, then the finite sequence {$c_k$} is saw-like." But I don't know how to prove the other direction. I think I need to construct a polynomial $f$ then prove that it satisfies all the properties. But I don't know how to construct it? Use Lagrange interpolation formula? Thanks!