\begin{align*} f(x) & = a \tan bx \\ f\left({\pi \over 2}\right) & = a \tan \left[b \left({\pi \over 2}\right)\right] \\ 0 & = a \tan {{\pi \over 2}b} \\ {0 \over a} & = \tan {{\pi \over 2}b} \\ 0 & = \tan {{\pi \over 2}b} \\ \\ \tan x & = 0 \text{ when } x = \pi, 2\pi, 3 \pi, ... \\ \\ \therefore {\pi \over 2}b & = \pi, 2\pi, 3 \pi, ... \\ b & = 2, 4, 6, ... \\ \\ \text{Smallest value of } b & = 2 \end{align*}

\begin{align*} f(x) & = a \tan bx \\ & = a \tan 2x \\ \\ f'(x) & = a (2)(\sec^2 2x) \phantom{000000} \left[ {d \over dx}[ \tan f(x) ] = f'(x) \sec^2 f(x) \right] \\ & = 2a \sec^2 2x \\ \\ f'\left({\pi \over 8}\right) & = 2a \sec^2 \left[2 \left({\pi \over 8}\right) \right] \\ 12 & = 2a \sec^2 {\pi \over 4} \\ 12 & = {2a \over \cos^2 {\pi \over 4}} \\ 12 & = {2a \over {1 \over 2}} \\ {1 \over 2}(12) & = 2a \\ 6 & = 2a \\ {6 \over 2} & = a \\ 3 & = a \end{align*}