In each of the following cases find a function with continuous second derivative
satisfying the given conditions.
-
for all
,
, and
.
-
for all
,
, and
.
-
for all
,
, and
for all
.
-
for all
,
, and
for all
.
- There can be no function meeting all of these conditions since
implies
is an increasing function (since its derivative,
, is positive). But then
contradicts that
is increasing.
- Let
. Then
Furthermore,
for all
.
- There can be no function meeting all of these conditions. Again,
for all
implies that
is increasing for all
. Therefore,
implies
for all
. Then, by the mean-value theorem, we know that for any
there exists some
such that
Now, choose
. Then,
, contradicting that
for all
.
- We’ll define
piecewise as follows
Then, we can take the derivative of each piece (and see that they are equal, so the derivative is defined at
)
Taking the derivative again we find
Thus,
for all
and
. Furthermore, for
we have