A)
is continuous, so
is also continuous. This means if we were to integrate
, the same constant of integration would apply across its entire domain. Over
, we have
. This means that
For
to be continuous, we need the limit as
to match
. This means we must have
Now, over
, we have
, so
, which means
.
b) Integrating over [1, 3] is easy; it's just the area of a 2x2 square. So,
c)
is increasing when
, and concave upward when
, i.e. when
is also increasing.
We have
over the intervals
and
. We can additionally see that
only on
and
.
d) Inflection points occur when
, and at such a point, to either side the sign of the second derivative
changes. We see this happening at
, for which
, and to the left of
we have
decreasing, then increasing along the other side.
We also have
along the interval
, but even if we were to allow an entire interval as a "site of inflection", we can see that
to either side, so concavity would not change.
