Question 1 (20 marks)
(1) We have three matrices given as follows:
$\begin{pmatrix} 2 & 2 \\ 0 & -1 \end{pmatrix}$, $\begin{pmatrix} 2 & 2 \\ -1 & -2 \end{pmatrix}$, $\begin{pmatrix} 1 & 1 \\ 0 & 2 \end{pmatrix}$.
Another $2 \times 2$ matrices X and Y satisfy $BX = D + X$ and $YB = C$. Compute X and
Y (show necessary steps).
(2) Calculate, step-by-step, the first and second derivatives and the maximum value of
the following functions. You may assume that the maximum points exist in all cases.
(a) $f(x) = -2x^2 + 10x - 8$, $x \in [3, 5]$
(b) $f(x) = 10 - \frac{2}{x+2}$, $x \in [5, 10]$
(c) $f(x) = -e^{-6x} + 6e^x$, $x \in (-\infty, \infty)$
(d) $f(x) = -ln(x^2 + 2x + 2)$, $x \in (-\infty, \infty)$.
