Forget about face recognition, James. The latest SPECTRE internet password requires you to combine functions in the infinite dimensional vector space of functions to obtain a " 0 -key" to crack into their evil web-portal. The pass key app here shows the functions \( e^{x}, e^{-x}, \cosh x \), and \( \sinh x \) together with the linear key combination
\[
f(x)=a e^{x}+b e^{-x}+c \cosh x+d \sinh x .
\]
Your mission, if you choose to accept it, is to find some combination of coefficients that will:
i) ensure that the function
\[
f(x)=a e^{x}+b e^{-x}+c \cosh x+d \sinh x
\]
is the zero function (or at least very close to it), and
ii) at least one of \( a, b, c, d \) is exactly equal to 1 .
Your solution James? Clearly the answer, or at least one answer, M, is given by the vector
\[
\left(\begin{array}{l}
a \\
b \\
c \\
d
\end{array}\right)=\square \text { ?. }
\]
Recall: the Maple notation for the vector \( \left(\begin{array}{l}1 \\ 2 \\ 3 \\ 4\end{array}\right) \) is \( \langle 1,2,3,4> \).
