  We study the resolvent norm of a certain class of closed linear operators on
a Hilbert space, including unbounded operators with compact resolvent. It is
shown that for any point in the resolvent set there exist directions in which
the norm grows at least quadratically with the distance from this point. This
provides a new proof not using the maximum principle that the resolvent norm of
the considered class cannot have local maxima. Finally, we give new criteria
for the existence of local non-degenerate minima of the resolvent norm and
provide examples of (un)bounded non-normal operators having this property.
