本题目来源于试卷: Maclaurin Series，类别为 IB数学
[问答题]
The function f is p a8mx 6dhc x2h 5czogtf+8l,8j or8ut s)m7n 6n(6lhudefined by $ f(x)=(\arccos x)^{2},-1 \leq x \leq 1 $.
1. Show that $ f^{\prime}(0)=-\pi $
The function f satisfies the equation
$\left(1-x^{2}\right) f^{\prime \prime}(x)-x f^{\prime}(x)=2$ .
2. By differentiating the above equation twice, show that
$\left(1-x^{2}\right) f^{(4)}-5 x f^{\prime \prime \prime}(x)=4 f^{\prime \prime}(x)$
where f^{(n)}(x) denotes the n th derivative of f(x) .
3. Hence show the Maclaurin series for f(x) up to and including the term in $x^{4}$ is $\frac{\pi^{2}}{4}-\pi x+x^{2}-\frac{\pi}{6} x^{3}+\frac{x^{4}}{3}$ .
4. Use this series approximation for f(x) with $x=\frac{1}{2}$ to find an approximate value for $25 \pi-\frac{20}{3} \pi^{2}$ .
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