Here we're given F prime of X,
or the first derivative function.
We're asked to determine
if the graph of F of X
at X equals one is
increasing or decreasing,
as well as whether it's
concave up or concave down.
We can determine if the
function is increasing
or decreasing by the sign
of the first derivative.
We can determine if the
function is concave up or down
by the sign on the second derivative.
So for a quick review,
if the first derivative is positive,
the function is increasing.
If the first derivative is negative,
the function is decreasing,
and if the second derivative is positive,
the function is concave up,
and if the second derivative is negative,
the function is concave down.
We'll notice here we're
given the first derivative,
so if we evaluate the derivative
function at X equals one,
we can determine if the
function is increasing
or decreasing at this value.
So F prime of one
would be equal to negative three
times the quantity one plus one
times the quantity one minus two,
which would be equal to negative three
times two, times negative one,
which equals positive six,
which is greater than zero,
which means at X equals one,
the function F of X is increasing.
Now, to determine whether
the function is concave up
or concave down at X equals one,
we do have to find the second derivative.
We're given the first
derivative, F prime of X
is equal to negative three
times the quantity X plus one
times the quantity X minus two.
So we could find the second derivative
by applying the product rule,
but let's go ahead and multiply this out,
and then find the second derivative.
So here we'd have negative three times,
multiplying this out, we'd have X squared,
and then minus two X plus one X.
That's minus X,
and then minus two.
We'll go ahead and distribute,
so we have negative three X squared,
plus three X, plus six.
So this is the first derivative,
so the second derivative would be
the derivative of the first derivative,
which would be negative six X
plus three.
So now we'll evaluate
our second derivative
at X equals one to determine
whether the function is
concave up or concave down
at X equals one.
So F double prime of one
is equal to negative six
times one plus three,
which is equal to negative three,
which is less than zero,
which means that at X equals one,
the function is concave down.
So the function is
increasing and concave down
at X equals one,
which means our answer is B.
Let's also look at this graphically.
In blue we have the graph
of our derivative function,
and in red we have the graph of our
second derivative function,
and again, where the first
derivative is positive,
the function is increasing.
So notice how our function
F of X would be increasing
on the interval from
negative one to positive two.
The function F of X would be decreasing
on the interval from negative infinity
to negative one, this interval here,
and also decreasing on the
interval from two to infinity.
One thing we should notice here is
if the function changes from
decreasing to increasing
at X equals negative one,
and therefore we'd have a relative minimum
at that location,
and then X equals two,
the function changes from
increasing to decreasing,
so we'd have a relative
maximum at X equals two.
And now looking at our
second derivative function,
notice how the second
derivative is positive
on the interval from negative infinity
to positive 1/2,
which means on this interval,
the function would be concave up,
and then on the interval from 1/2
to positive infinity here,
the second derivative is negative,
and therefore the function
would be concave down
on this interval.
Notice how the function teaches concavity
at X equals negative 1/2,
and therefore at X equals 1/2,
we'd have a point of inflection.
And finally, to verify our
answer on the previous slide,
we were describing our function F of X
at X equals one.
At X equals one, notice
how the first derivative
graphed here in blue is positive,
and that's why F of X is increasing.
At X equals one, the second
derivative is negative here,
and that's why at X equals one,
our function is concave down.
So just keep in mind
these descriptions here
are describing F of X
based upon the function values
of F prime of X,
graphed here in blue,
and F double prime of X,
graphed here in red.
I hope you found this helpful.
