10\\
\begin{tikzpicture}[x=1.8cm, y=1.2cm, >=latex]

    % Draw axes
    \draw[->, semithick] (-0.5, 0) -- (4.4, 0) node[right] {$x$};
    \draw[->, semithick] (0, -2.6) -- (0, 6) node[left] {f($x$)};

    % Draw x-axis ticks
    \foreach \x in {1, 2, 3, 4} {
        \draw[semithick] (\x, 0) -- (\x, -0.08) node[below] {\x};
    }
    % Origin label
    \node[below] at (0, -0.08) {0};

    % Draw the function curve
    \begin{scope}
        % Clip the area to cleanly cut off the lines heading towards infinity
        \clip (-0.5, -2.6) rectangle (4.2, 6);
        
        % Left branch of the curve (before the x = 0.5 asymptote)
        \draw[thick, samples=200, domain=-0.35:0.47] plot (\x, {exp(\x)/(4*\x*\x - 1) + 2});
        
        % Right branch of the curve (after the x = 0.5 asymptote)
        \draw[thick, samples=200, domain=0.53:4.2] plot (\x, {exp(\x)/(4*\x*\x - 1) + 2});
    \end{scope}

\end{tikzpicture}

The diagram shows part of the curve $\mathrm { f } ( x ) = \frac { \mathrm { e } ^ { x } } { 4 x ^ { 2 } - 1 } + 2$. The equation $\mathrm { f } ( x ) = 0$ has a positive root $\alpha$ close to $x = 0.3$.
\begin{enumerate}[label=(\alph*)]
\item Explain why using the sign change method with $x = 0$ and $x = 1$ will fail to locate $\alpha$.
\item Show that the equation $\mathrm { f } ( x ) = 0$ can be written as $x = \frac { 1 } { 4 } \sqrt { \left( 4 - 2 \mathrm { e } ^ { x } \right) }$.
\item Use the iterative formula $x _ { n + 1 } = \frac { 1 } { 4 } \sqrt { \left( 4 - 2 \mathrm { e } ^ { x _ { n } } \right) }$ with a starting value of $x _ { 1 } = 0.3$ to find the value of $\alpha$ correct to $\mathbf { 4 }$ significant figures, showing the result of each iteration.
\item An alternative iterative formula is $x _ { n + 1 } = \mathrm { F } \left( x _ { n } \right)$, where $\mathrm { F } \left( x _ { n } \right) = \ln \left( 2 - 8 x _ { n } ^ { 2 } \right)$.

By considering $\mathrm { F } ^ { \prime } ( 0.3 )$ explain why this iterative formula will not find $\alpha$.
\end{enumerate}

\hfill \mbox{\textit{OCR H240/01 2023 Q10 [9]}}