GRAPH of secular antiderivative $g_{\mu}(\cdot)$ in easy case.
\[
g_{\mu}(\lambda) = \lambda -
\sum_{i=1}^{n-1}\left(\frac{\hat{\eta}_i^2}{\gamma_i-
\lambda}\right),
\]
$g_{\mu}(\cdot)$ possesses a singularity at each eigenvalue
$\gamma_i,~i=1,2,\ldots,n-1$.

$g_{\mu}(\lambda) \rightarrow \infty$ as
$\lambda \downarrow \gamma_i$, while
$g_{\mu}(\lambda) \rightarrow -\infty$ as $\lambda \uparrow \gamma_i$,
for each $i=1,2,\ldots,n-1$.  

Let $\gamma_{i+1} < \gamma_i$.
at least one root of
$g_{\mu}(\cdot)$ in $(\gamma_{i+1},\gamma_i)$;
$g_{\mu}'''(\lambda) < 0$,  and
consequently $g_{\mu}''(\lambda)$ is monotone decreasing in this
interval.  

\newpage
\[
g_{\mu}(\lambda) = \lambda -
\sum_{i=1}^{n-1}\left(\frac{\hat{\eta}_i^2}{\gamma_i-
\lambda}\right),
\]
$g_{\mu}(\cdot)$ has at most one point of
inflection
on $(\gamma_{i+1},\gamma_i)$, which is possibly also a critical point.
  Should there be a point of inflection in
$(\gamma_{i+1},\gamma_i)$, then on that interval
$g_{\mu}(\cdot)$ is strictly convex to the left of
this point, and strictly concave to the right of it.
Hence $g_{\mu}(\cdot)$ has either zero, one, or two critical points on
  $(\gamma_{i+1},\gamma_i)$, with the possibility of only one critical
point being accounted for by the existence of an inflection which is also
critical.  

$g_{\mu}(\cdot)$
is strictly convex on the semi-infinite
interval $(\gamma_1,\infty)$, while we have strict concavity on
the other semi-infinite interval, namely $(-\infty,\gamma_{n-1})$.
\newpage
\[
g_{\mu}(\lambda) = \lambda -
\sum_{i=1}^{n-1}\left(\frac{\hat{\eta}_i^2}{\gamma_i-
\lambda}\right)
\]

unique
critical point, namely $\lambda_1$, on $(\gamma_1,\infty)$

unique
critical
point, namely $\lambda_k$, on $(-\infty,\gamma_{n-1})$.

set of
critical points of the secular antiderivative function $g_{\mu}(\cdot)$ is
$\Lambda$.

\[
\mu_1 = g_{\mu}(\lambda_1) > \mu_2=g_{\mu}(\lambda_2) > \ldots >
\mu_k=g_{\mu}(\lambda_k),
\]
 \[ \mu_i = \mu(x^{\lambda_i}), ~i=1,2,\ldots,k \]
for the {\em stationary values} of $\mu(\cdot)$ on $S_{n-1}$.
