Based on a previous question, I had the following conjecture and was wondering if anyone knew how to prove it or find a counterexample.
Consider the polynomial $ p(t)=c\frac{(t-x_0)(t-x_1)\cdots(t-x_n)}{(x-x_0)(x-x_1)\cdots(x-x_n)}$
where $x_0,x_1,\ldots,x_n$, and $x$ are distinct and all lie in the interval $[a,b]$, and $c\neq 0$. The polynomial $p(t)$ is the lagrange interpolation polynomial of degree $n+1$ satisfying $p(x_i) = 0$ for $i=0,\ldots,n$ and $p(x)=c$. Its $(n+1)$-th derivative is the constant $p^{(n+1)}(t)=\frac{c(n+1)!}{(x-x_0)(x-x_1)\cdots(x-x_n)}$
Conjecture: suppose $f(t)\in C^{(n+1)}[a,b]$ and satisfies $f(x_i) = 0$ for $i=0,\ldots,n$ and $f^{(n+1)}(t)>p^{(n+1)}(t)$ for all $t\in [a,b]$ or $f^{(n+1)}(t) for all $t\in [a,b]$. Prove that $f(x) \neq p(x)=c$. Or find a counter example to the conjecture