  We study the number of local maxima with given radial derivative of spherical
mixed $p$-spin models and prove that the second moment matches the square of
the first moment on exponential scale for arbitrary mixtures and any radial
derivative. This is surprising, since for the number of local maxima with given
radial derivative and given energy the corresponding result is only true for
specific mixtures [Sub17; BSZ20]. We use standard Kac-Rice computations to
derive formulas for the first and second moment at exponential scale, and then
find a remarkable analytic argument that shows that the second moment formula
is bounded by twice the first moment formula in this general setting. This also
leads to a new proof of a central inequality used to prove concentration of the
number critical points of pure $p$-spin models of given energy in [Sub17] and
removes the need for the computer assisted argument used in that paper for $3
\leq p \leq 10$.
