2 For complex numbers $a, b, c$, consider the polynomial $f(z) = az^2 + bz + c$. Let $i$ be the imaginary unit.
[(1)] Let $\alpha$, $\beta$, $\gamma$ be complex numbers. When $f(0) = \alpha$, $f(1) = \beta$, $f(i) = \gamma$ hold, express $a$, $b$, $c$ in terms of $\alpha$, $\beta$, $\gamma$ respectively.
[(2)] When $f(0)$, $f(1)$, $f(i)$ are all real numbers satisfying $1 \leq f(0),\, f(1),\, f(i) \leq 2$, illustrate the range of values that $f(2)$ can take on the complex plane.
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Let $f(x) = \dfrac{x}{x^2+3}$, and let $C$ be the graph of $y = f(x)$. Let $l: y = g(x)$ be the tangent line to $C$ at the point $\mathrm{A}(1,\ f(1))$.
[(1)] Show that there exists exactly one point on $C$ that is common to $C$ and $l$ and is different from $\mathrm{A}$, and find the $x$-coordinate of that point.
[(2)] Let $\alpha$ be the $x$-coordinate of the common point found in (1). Compute the definite integral $\displaystyle\int_{\alpha}^{1} \{f(x) - g(x)\}^2\, dx$.
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\textbf{2}
For complex numbers $a, b, c$, consider the polynomial $f(z) = az^2 + bz + c$. Let $i$ be the imaginary unit.
\begin{enumerate}
\item[(1)] Let $\alpha$, $\beta$, $\gamma$ be complex numbers. When $f(0) = \alpha$, $f(1) = \beta$, $f(i) = \gamma$ hold, express $a$, $b$, $c$ in terms of $\alpha$, $\beta$, $\gamma$ respectively.
\item[(2)] When $f(0)$, $f(1)$, $f(i)$ are all real numbers satisfying $1 \leq f(0),\, f(1),\, f(i) \leq 2$, illustrate the range of values that $f(2)$ can take on the complex plane.
\end{enumerate}
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\noindent\boxed{3}
\medskip
Let $f(x) = \dfrac{x}{x^2+3}$, and let $C$ be the graph of $y = f(x)$. Let $l: y = g(x)$ be the tangent line to $C$ at the point $\mathrm{A}(1,\ f(1))$.
\begin{enumerate}
\item[(1)] Show that there exists exactly one point on $C$ that is common to $C$ and $l$ and is different from $\mathrm{A}$, and find the $x$-coordinate of that point.
\item[(2)] Let $\alpha$ be the $x$-coordinate of the common point found in (1). Compute the definite integral $\displaystyle\int_{\alpha}^{1} \{f(x) - g(x)\}^2\, dx$.
\end{enumerate}
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