We prove two results concerning extendible cardinals. First, we show that the first
C
(
n
)
C^{(n)}
-extendible cardinal is strictly greater than the first
C
(
n
)
C^{(n)}
-supercompact. This answers a question posed by Bagaria in [Arch. Math. Logic 51 (2012), pp. 213–240]. Second, assuming the existence of strong enough large cardinals, we prove the consistency of the following: there are cardinals
κ
>
λ
\kappa >\lambda
such that
λ
\lambda
is singular of countable cofinality,
κ
\kappa
is
>
λ
{>}\lambda
-extendible (but not
λ
\lambda
-extendible) and
(
λ
+
)
H
O
D
>
λ
+
(\lambda ^+)^{\mathrm {HOD}}>\lambda ^+
.