Intermediate value theorem.
Extrema occur at points where
, with maxima occurring at
if the derivative is positive to the left of
and negative to the right of
, and minima in the opposite case.
So suppose you take two values
. If it turns out that
and
, then the IVT guarantees the existence of some
such that
.
Choosing arbitrary values of
won't guarantee that exactly one such
exists, though. The function could easily oscillate several more times between
and
, intersecting the x-axis more than once, for example. This is where your suspicion can be applied. Knowing that
for
(approximately 1.57, 4.71, 7.85, respectively), you can use these values as reference points for computing the sign of the derivative.
When
, you have
. You know that
for
, and that as
,
. This means there must be some
such that
, and in particular, this value of
is the site of a relative maximum.
You can use similar arguments to determine what happens at the other two suspected critical points.