The the proof of Lemma 1 that can be found in the book Calculus of variations by I. M. Gelfand and S. V. Fomin, as shown in the following image, involves the definition of the function $h(x)=(x-x_1)(x_2-x)$ . However, if we substitute the hypothesis $h(x)\in C(a,b)$ by $h(x)\in D_n(a,b)$ where $D_n(a,b)$ is the set of all continuous functions with continuous derivatives up to order $n$ in $(a,b)$, the following Remark indicates that we would need to define another function $h(x)$ to prove the lemma, namely:
$h(x)=[(x-x_1)(x_2-x)]^{n+1}$
Why is this necessary? Wouldn't the same function $h(x)$ defined before serve the purpose?
The function $h$ is a piecewise function. Indeed, the first derivative of the $h$ function defined in the proof would not be continuous, it is not even defined everywhere. (you will have problem finding $h'(x_1)$ or $h'(x_2)$.)
By taking the $n+1$ power of the preceding function, this will fix that problem and you will obtain a $n$ times derivable function with the $n$-th derivative being continuous. Then the $h$ defined in the remark is indeed in $D_n[a,b]$ while the first one was not.