Answer
$(a).$ semi-annually compounded $r=5.125\%$ provides the best Investment
Work Step by Step
The formula for continuously compounded Interest is, $A(t)=Pe^{rt}$ whereas, $P=Initial-Investment$, $r=rate$, $e=natural-logarithm's-constant-base$ and $t=time$. And formula for periodically(annualy,semi-annualy,quarterly,monthly,daily and so on) compounded Interest is, $A(t)=P\left(1+\frac{r}{n} \right)^{nt}$ whereas, $P=Intial-Investment,$ $rate,$ $n=number-of-times-it-compounds,$ and $time$.
(a) is for compounded semi-annualy, (b) is for compounded continuously.
Thus, The Table is as follows:
$\begin{array}{ll}
(a).r=5.125\% & (b).r=5\% & \\
P(1+\frac{r}{n})^{nt} & Pe^{rt} \\
P(1+\frac{0.05125}{2})^{2t} & Pe^{0.05t} \\
P(1.05191)^t & P(1.05127)^t
\end{array}$
$P(1.05191)^t\gt P(1.05127)^t$. Therefore, $(a).r=5.125\%$ provides the best Investment
