  We study the quantity $$
\tau_{n,k}:=\frac{|T_n^{(k)}(\omega_{n,k})|}{T_n^{(k)}(1)}\,, $$ where $T_n$ is
the Chebyshev polynomial of degree $n$, and $\omega_{n,k}$ is the rightmost
zero of $T_n^{(k+1)}$.
  Since the absolute values of the local maxima of $T_n^{(k)}$ increase
monotonically towards the end-points of $[-1,1]$, the value $\tau_{n,k}$ shows
how small is the largest critical value of $\,T_n^{(k)}\,$ relative to its
global maximum $\,T_n^{(k)}(1)$.
  In this paper, we improve and extend earlier estimates by
Erd\H{o}s--Szeg\H{o}, Eriksson and Nikolov in several directions.
  Firstly, we show that the sequence $\,\{\tau_{n,k}\}_{n=k+2}^{\infty}$ is
monotonically decreasing in $n$, hence derive several sharp estimates, in
particular $$ \tau_{n,k} \le \begin{cases} \tau_{k+4,k} =
\frac{1}{2k+1}\,\frac{3}{k+3}\,, & n \ge k+4\, \tau_{k+6,k} = \frac{1}{2k+1}\,
(\frac{5}{k+5})^2 \beta_k\,,
  & n \ge k+6\,, \end{cases} $$ where $\beta_k < \frac{2+\sqrt{10}}{5} \approx
1.032$.
  We also obtain an upper bound which is uniform in $n$ and $k$, and that
implies in particular $$
  \tau_{n,k} \approx \big(\frac{2}{e}\big)^k, \quad n \ge k^{3/2}; \qquad
  \tau_{n,n-m} \approx \big(\frac{em}{2}\big)^{m/2} n^{-m/2}; \qquad
  \tau_{n,n/2} \approx \big(\frac{4}{\sqrt{27}}\big)^{n/2}. $$ Finally, we
derive the exact asymptotic formulae for the quantities $$ \tau_k^{*} :=
\lim_{n\to\infty}\tau_{n,k} \quad \mbox{ and }\quad \tau_m^{**} :=
\lim_{n\to\infty} n^{m/2} \tau_{n,n-m}\,, $$ which show that our upper bounds
for $\tau_{n,k}$ and $\tau_{n,n-m}$ are asymptotically correct with respect to
the exponential terms given above.
