  We develop a rigorous, semi-analytical method for maximizing any $b\to
c\tau\nu$ observable in the full 20-real-dimensional parameter space of the
dimension 6 effective Hamiltonian, given some fixed values of $R_{D^{(*)}}$. We
apply our method to find the maximum allowed values of $F^L_{D^*}$ and
$R_{J/\psi}$, two observables which have both come out higher than their SM
predictions in recent measurements by the Belle and LHCb collaborations. While
the measurements still have large error bars, they add to the existing
$R_{D^{(*)}}$ anomaly, and it is worthwhile to consider NP explanations. It has
been shown that none of the existing, minimal models in the literature can
explain the observed values of $F^L_{D^*}$ and $R_{J/\psi}$. Using our method,
we will generalize beyond the minimal models and show that there is no
combination of dimension 6 Wilson operators that can come within $1\sigma$ of
the observed $R_{J/\psi}$ value. By contrast, we will show that the observed
value of $F^L_{D^*}$ can be achieved, but only with sizable contributions from
tensor and mixed-chirality vector Wilson coefficients.
