The function
is defined as:

To find the stationary points of
, we need to find the values of
where the derivative of
is equal to zero.
First, let's find the derivative of
:

To find the stationary points, we set
and solve for
:

We can factor the quadratic equation as follows:

Now, we solve for
by factoring further:

This gives us two solutions:
and
.
So, the stationary points of
are
and
.
To determine the intervals where
is increasing, we need to analyze the sign of the derivative
in different intervals. We can use the values of
,
, and any other value between them.
For
, we choose
as a test point:

For
, we choose
as a test point:

For
, we choose
as a test point:

From the above analysis, we can conclude that
is increasing in the intervals
and
.
To find the inflection point of
, we need to determine where the concavity changes. This occurs when the second derivative of
changes sign.
The second derivative of
is:

To find the inflection point, we set
and solve for
:



Therefore, the inflection point of
is
.