本题目来源于试卷: Differential Calculus，类别为 IB数学
[问答题]
Consider the functiozsu q 18rouhi; ih-;cx3jd7* xn 2os bz8 +hur1/e(co3(fylo.* jqaofz $f(x)=\cos (p \arccos x),$-1 \lt x \lt 1$ and p \in \mathbb{R}$ .
1. Show that $f^{\prime}(0)=p \sin \left(\frac{p \pi}{2}\right) $.
The function f and its derivative satisfy
$\left(1-x^{2}\right) f^{(n+2)}(x)=(2 n+1) x f^{(n+1)}(x)+\left(n^{2}-p^{2}\right) f^{(n)}(x), \quad n \in \mathbb{N}$
where $f^{(n)}(x)$ denotes the n th derivative of f(x) and $f^{(0)}(x)$ is f(x) .
2. Show that $ f^{(n+2)}(0)=\left(n^{2}-p^{2}\right) f^{(n)}(0)$ .
3. For $p \in \mathbb{R} \backslash\{ \pm 1, \pm 2, \pm 3\}$ , show that the Maclaurin series for f(x) , up to and including the x^{4} term, is
$\begin{array}{l}
\cos \left(\frac{p \pi}{2}\right)+p \sin \left(\frac{p \pi}{2}\right) x-\frac{p^{2} \cos \left(\frac{p \pi}{2}\right)}{2} x^{2} \\
+ \frac{\left(1-p^{2}\right) p \sin \left(\frac{p \pi}{2}\right)}{6} x^{3}-\frac{\left(4-p^{2}\right) p^{2} \cos \left(\frac{p \pi}{2}\right)}{24} x^{4}
\end{array}$
4. Hence or otherwise, find $\lim _{x \rightarrow 0} \frac{\cos (p \arccos (x))}{x}$ where p is an odd integer.
5. If p is an integer, prove the Maclaurin series for f(x) is a polynomial of
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