Computer Science applicants should turn to page 14. (i) The graph $y = f ( x )$ of a certain function has been plotted below. [Figure] On the next three pairs of axes (A), (B), (C) are graphs of $$y = f ( - x ) , \quad f ( x - 1 ) , \quad - f ( x )$$ in some order. Say which axes correspond to which graphs. [Figure] (A) [Figure] (B) [Figure] (C) (ii) Sketch, on the axes opposite, graphs of both of the following functions $$y = 2 ^ { - x ^ { 2 } } \quad \text { and } \quad y = 2 ^ { 2 x - x ^ { 2 } }$$ Carefully label any stationary points. (iii) Let $c$ be a real number and define the following integral $$I ( c ) = \int _ { 0 } ^ { 1 } 2 ^ { - ( x - c ) ^ { 2 } } \mathrm {~d} x$$ State the value(s) of $c$ for which $I ( c )$ is largest. Briefly explain your reasoning. [Note you are not being asked to calculate this maximum value.] [Figure]
(i) [3 marks] (A) is $- f ( x ) ; \quad ( \mathrm { B } )$ is $f ( - x ) ; \quad ( \mathrm { C } )$ is $f ( x - 1 )$. [0pt]
\section*{3. For APPLICANTS IN $\left\{ \begin{array} { l } \text { MATHEMATICS } \\ \text { MATHEMATICS \& STATISTICS } \\ \text { MATHEMATICS \& PHILOSOPHY } \\ \text { MATHEMATICS \& COMPUTER SCIENCE } \end{array} \right\}$ ONLY.}
Computer Science applicants should turn to page 14.\\
(i) The graph $y = f ( x )$ of a certain function has been plotted below.\\
\includegraphics[max width=\textwidth, alt={}, center]{2b5c0f85-f6ae-4a52-aaf5-c1ed5f1a7e7e-10_384_575_708_737}
On the next three pairs of axes (A), (B), (C) are graphs of
$$y = f ( - x ) , \quad f ( x - 1 ) , \quad - f ( x )$$
in some order. Say which axes correspond to which graphs.\\
\includegraphics[max width=\textwidth, alt={}, center]{2b5c0f85-f6ae-4a52-aaf5-c1ed5f1a7e7e-10_384_575_1414_223}\\
(A)\\
\includegraphics[max width=\textwidth, alt={}, center]{2b5c0f85-f6ae-4a52-aaf5-c1ed5f1a7e7e-10_378_572_1416_822}\\
(B)\\
\includegraphics[max width=\textwidth, alt={}, center]{2b5c0f85-f6ae-4a52-aaf5-c1ed5f1a7e7e-10_382_579_1416_1411}\\
(C)\\
(ii) Sketch, on the axes opposite, graphs of both of the following functions
$$y = 2 ^ { - x ^ { 2 } } \quad \text { and } \quad y = 2 ^ { 2 x - x ^ { 2 } }$$
Carefully label any stationary points.\\
(iii) Let $c$ be a real number and define the following integral
$$I ( c ) = \int _ { 0 } ^ { 1 } 2 ^ { - ( x - c ) ^ { 2 } } \mathrm {~d} x$$
State the value(s) of $c$ for which $I ( c )$ is largest. Briefly explain your reasoning. [Note you are not being asked to calculate this maximum value.]\\
\includegraphics[max width=\textwidth, alt={}, center]{2b5c0f85-f6ae-4a52-aaf5-c1ed5f1a7e7e-11_750_1390_178_283}\\