3.55: Exponential derivative; GP and trig GP

The equation of a curve is $y=\dfrac{\mathrm{e}^{-3x+2}}{x+1}$, where $x>-1$. Show that
$$\dfrac{\mathrm{d}y}{\mathrm{d}x}=\dfrac{\mathrm{e}^{-3x+2}(Ax+B)}{(x+1)^{2}},$$
where $A$ and $B$ are integers to be found.

Hence show there is only one stationary point and find its exact coordinates.

A GP has $3$rd term $6$ and $8$th term $1458$. Find $r$ and $a$.

A geometric progression has first three terms $\mathrm{cos}\,\theta$, $\dfrac{1}{2}\mathrm{cos}^{2}\theta$, $\dfrac{1}{4}\mathrm{cos}^{3}\theta$ for $-90^{\circ}<\theta<90^{\circ}$. Find the values of $\theta$ for which the progression has a sum to infinity.