It is well known that the derivative of the Minkowski function
?
(
x
)
?(x)
(whenever exists) may take only two values:
0
0
and
+
∞
+\infty
. Let
E
n
\textbf {E}_n
be the set of irrational numbers on the interval
[
0
;
1
]
[0; 1]
whose partial quotients (related to the continued fraction expansion) do not exceed
n
n
. It is also known that the quantity
?
′
(
x
)
?’(x)
at a point
x
=
[
0
;
a
1
,
a
2
,
…
,
a
t
,
…
]
x=[0;a_1,a_2,\dots ,a_t,\dots ]
is linked with the limit behavior of the arithmetic means
(
a
1
+
a
2
+
⋯
+
a
t
)
/
t
(a_1+a_2+\dots +a_t)/t
. In particular, A. Dushistova, I. Kan, and N. Moshchevitin showed that if
x
∈
E
n
x\in \textbf {E}_n
satisfies
a
1
+
a
2
+
⋯
+
a
t
>
(
κ
1
(
n
)
−
ε
)
t
a_1+a_2+\dots +a_t>(\kappa ^{(n)}_1-\varepsilon ) t
, where
ε
>
0
\varepsilon >0
and
κ
1
(
n
)
\kappa ^{(n)}_1
is a certain explicit constant, then
?
′
(
x
)
=
+
∞
?’(x)=+\infty
. They also showed that the quantity
κ
1
(
n
)
\kappa ^{(n)}_1
cannot be increased. In the present paper, a dual problem is treated: how small may the quantity
a
1
+
a
2
+
⋯
+
a
t
−
κ
1
(
n
)
t
a_1+a_2+\dots +a_t-\kappa ^{(n)}_1 t
be if
?
′
(
x
)
=
0
?’(x)=0
? Optimal estimates in this problem are found.