  Let M be a closed compact n-dimensional manifold with n odd. We calculate the
first and second variations of the zeta-regularized determinants
det^\prime\Lambda and det L as the metric on M varies, where \Delta denotes the
Laplacian on functions and L denotes the conformal Laplacian. We see that the
behavior of these functionals denotes the conformal Laplacian. We see that the
behavior of these functionals depends on the dimension. Indeed, every critical
metric for (-1)^{(n-1)/2}det^\prime\Lambda or (-1)^{(n-1}/2}| det L| has finite
index. Consequently there are no local maxima if n=4m+1 and no local minima if
n=4m+3. We show that the standard 3-sphere is a local maximum for
det^\prime\Lambda while the standard (4m-3)-sphere with m=1,2,...,4, is a
saddle point. By contrast, for all odd n, the standard n-sphere is a local
extremal for det L.
  An important tool in our work is the canonical trace on odd class operators
in odd dimensions. This trace is related to the determinant by the formula det
Q = TR log Q, and we prove some basic results on how to calculate the trace.
