Let $\alpha$ be a positive real number. Define the function $f(\theta)$ of $\theta$ for $0 \leq \theta \leq \pi$ as the square of the distance AP between the two points $\mathrm{A}(-\alpha,\ -3)$ and $\mathrm{P}(\theta + \sin\theta,\ \cos\theta)$ in the coordinate plane.
[(1)] Show that there exists exactly one $\theta$ in the range $0 < \theta < \pi$ such that $f'(\theta) = 0$.
[(2)] Find the range of $\alpha$ such that the following holds: [6pt] The function $f(\theta)$ of $\theta$ for $0 \leq \theta \leq \pi$ attains its maximum at some point in the interval $0 < \theta < \dfrac{\pi}{2}$.
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Let $\alpha$ be a positive real number. Define the function $f(\theta)$ of $\theta$ for $0 \leq \theta \leq \pi$ as the square of the distance AP between the two points $\mathrm{A}(-\alpha,\ -3)$ and $\mathrm{P}(\theta + \sin\theta,\ \cos\theta)$ in the coordinate plane.
\begin{enumerate}
\item[(1)] Show that there exists exactly one $\theta$ in the range $0 < \theta < \pi$ such that $f'(\theta) = 0$.
\item[(2)] Find the range of $\alpha$ such that the following holds:\\[6pt]
The function $f(\theta)$ of $\theta$ for $0 \leq \theta \leq \pi$ attains its maximum at some point in the interval $0 < \theta < \dfrac{\pi}{2}$.
\end{enumerate}
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