Yup that is correct.
So now, if you want to find where it is concaving up and down. Find the points of inflection, i.e. when
Then simply plug in a point inbetween the points of inflection to see if it's increasing or decreasing. Then you know if it's concaving up or down!
Yup,
so let us check
This of course becomes
which has the roots
and
EDIT-- I didn't factor correctly. What we have here is
This introduces the root
So we know at these points the graph has an inflextion point. What you need to do now is sub in a point inbetween these bounds and outside of them.
In other words, there are 4 parts of this graph. We seperate them into the bounds like
is part 1
is part 2
is part 3
is part 4
Sub in a value of x that falls within each of these parts into your second derivative. This will produce the concavity of all 3 intervals
You can't plug in 1 and -1 for part 2. We already know the function is 0 here! We want to know how the function behaves inbetween these bounds.
Also, I believe I messed up, we have 4 segments not 3! If you notice
Is the same as saying
This equation is also satisfied for
Therefore our segments become
Part 1
Part 2
Part 3
Part 4
My bad!
But the same thing applies here. We must plug numbers inbetween these points. So for example, plug in -100 for part 1, -.5 for part 2, .5 for part 3 and 100 for part 4