Some estimates of differentiable functions

Suppose that $x(t)\in C_{[a,b]}^{(n)}$ and has $n$ zeros at the points $a$ and $b$. It is shown that if $x^{(n)}(t)$ preserves sign on $[a,b]$, then
$$ |x(t)|\ge\frac{p_0}{(n-1)}\Bigl[\sup\limits_{\tau\in(a,b)}\frac{|x(\tau)|}{(\tau-a)^{p-1}(b-\tau)^{q-1}}\Bigr](t-a)^p(b-t)^q\quad(a<t<b), $$
where $p$ and $q$ are the multiplicities of the zeros of $x(t)$ at $a$ and $b$, respectively, and $p_0=\min\{p,q\}$. Two-sided estimates of the Green's function for a two-point interpolation problem for the operator $Lx\equiv x^{(n)}$ are established in the proof. As an application, new conditions for the solvability of de la Vallée Poussin's two-point boundary problems are obtained.