We will compute the derivative and show that it is always
. The derivative of
is given by the quotient rule to be
The denominator is clearly always positive as we know it is the square of some real number
, so in order to show that the entire derivative is always positive we need to show that the numerator is positive.
However,
and we know that, by the mean value theorem there is some
such that
since
. Since
is monotonically increasing it must be that
.
Thus,
and so the derivative of
is positive, which implies that
is monotonically increasing.
Since the denominator of
goes to zero as
we can employ L'Hopital's rule. Using this we get
Since the denominator is again zero we apply L'Hopital's rule again
An example of a function that is not differentiable, but where this limit exists is
The second derivative of this function does not exist at zero, as
however we can compute the above limit at zero by
A function
is said to be convex if
First we shall prove (
) that
monotonically increasing implies
is convex.
Assume
is monotonically increasing. Let
for
. The mean value theorem implies that
such that
. Similarly
such that
. Thus, since
is assumed to be monotonically increasing we have
Because
we know that
for some
. Moreover, we have
We note here that the direction of the inequality is preserved since we have
and
and
(*)
reducing yields
but since the initial choice of
was arbitrary (up to ordering) this holds for any
. We must still address the case corresponding to
. This is easy though.
For let
. Then by what we have proven already
now let
so it follows from the above that
Thus if
is monotonically increasing
for
,
Now we prove (
) that
convex implies
is monotonically increasing.
The proof of this direction follows similarly to the previous. Let's assume
is convex.
Then
for
(Note: we must have
by (*)) and
,
. Following the previous reasoning in reverse we see that
for
.
So in the limit
we have
and in the limit
we have
thus, combining these statements we have
so
is monotonically increasing as desired.
Now it only remains to prove that (assuming
exists for
)
is convex if and only if
for
.
is convex
is monotonically increasing by the previous proof. Moreover,
is monotonically increasing
(proved in class).
So we have shown that (assuming
exists for
)
is convex if and only if
for
as desired.