Hi Rohin
You seemed to have either misread or mistyped the question. We need to prove for \(1\le t \lt 2\), \(f_x(t)\) = \(_1p_x\)\(q_{x+1}\).
From part (a), we proved for \(0 \le t\lt 1\), \(f_x(t)\) = \(q_x\) using the assumption that deaths are uniformly distributed.
For \(1\le t\lt 2\), we have:
\(f_x(t)\) = \(S_x(1)\) \(f_{x+1}(t-1)\)
This is as a result of thinking of the pdf of \(T_x\) for \(1 \le t \lt 2\) as the probability of someone surviving from age \(x\) to age \(x+1\) multiplied by the probability of someone dying within the period from age \(x+1\) to \(x+2\).
= \(_1p_x\)\(q_{x+1}\)