  A study of $R_b$ and $R_c$ is presented in the context of a Two Higgs Doublet
Model (2HDM) with flavor changing scalar currents (FCSC). Implications of the
model for the $\rho$-parameter and for $b\to s\gamma$ are also considered. The
experimental data on $R_b$ places stringent constraints on the model
parameters. The configuration of the model needed to account for $R_b$ is found
to be irreconcilable with constraints from $b\to s\gamma$ and $\rho$. In
particular, if $R^{\rm exp}_b>R^{\sss{\rm SM}}_b$ persists then this version of
2HDM will be ruled out or require significant modifications. Noting that
aspects of the experimental analysis for $R_b$ and $R_c$ may be of some
concern, we also disregard $R^{\rm exp}_b$ and $R^{\rm exp}_c$ and give
predictions for these using constraints from $b\to s\gamma$ and $\rho$
parameter only. We emphasize the theoretical and experimental advantages of the
observable $R_{b+c}\equiv \Gamma(Z\to b\bar b\mbox{ or } c\bar
c)/\Gamma(Z\to\mbox{hadrons})$. We also stress the role of $R_\ell\equiv
\Gamma(Z\to\mbox{hadrons})/\Gamma(Z\to \ell^+\ell^-)$ in testing the Standard
Model (SM) despite its dependence on QCD corrections. Noting that in models
with FCNC the amplitude for $Z\to c\bar c$ receives a contribution which grows
with $m^2_t$, the importance and uniqueness of precision $Z\to c\bar c$
measurements for constraining flavor changing $t\bar c$ currents is
underscored.
