Which of the points labeled by letters in the graph of f viewable by
clicking the following link have:

http://photos1.blogger.com/img/20/1056/200/DSC03885.jpg

(a) F prime and F double prime nonzero and the same sign?
(b) At least two of F, F prime, and F double prime equal to zero?

I have the answer, I just need it written up and explained through a
step by step process. If the question doesn't make sense, click on the
picture, it's there put more clearer.

I'd like to help on this problem as well.  I'd like to keep the same
level of integrity as Researchers, so I am requesting that prior to
helping you further, please tell me the answer you currently have and
any approach(es) you have tried.  This will allow me to better isolate
your trouble, and it will also give me piece of mind that you are
interested in this problem for reasons other than directly copying my
comments into a graded assignment.

Thank you.

Well, let's think about this logically;

f is your function, and is represented by the black line.  It is equal
to the VERTICAL (Y) POSITION for a given value of x.

f' is the derivative of your function.  In other words, it is the
SLOPE at any point x.

f'' is the second derivative of your function.  It is effectively the
SLOPE OF THE SLOPE.

So, f' is positive whenever your slope is positive.  Remember that a
positive slope is like a / and a negative slope is like a \.

Regarding f'', it is positive whenever your line curves
counterclockwise, and vice versa.

So, from the beginning of the plot to A, your function is curving
counterclockwise and your slope is negative.

Between A and B, you're curving counterclockwise and your slope is positive.

Between B and C, you're curving clockwise and your slope is positive.

And you can figure out the rest on your own.

When your slope (f') is zero, that means you are at a local minumum or
maximum, so something like point A.

When your f'' is zero, that means you are at a point where your curve
switches from counterclockwise to clockwise (or vice versa).  This
point is called a "point of inflection".  Point B is an example of
this.

I don't want to give you the exact answer to this problem, because it
seems like a  homework problem and we are not supposed to just answer
those willy nilly.  Also, you won't learn anything if I just give you
the answer.  But, I do believe that by teaching you the method and how
to apply the reasoning, you will be able to get the answer on your own
and actually understand what's going on.

I got this problem wrong, but the teacher maked the correct answer on the test as: 

(a) B and E and (b) A and D. 

I believe it has something to do with f prime increasing or decreasing
and f double prime deals with if the point is concave up or down, but
i'm not possitive on how to put everything together. Also, F(x) deals
with where it's at on the plot.

Where i'm lost is, I didn't know what exactly the question is asking.
If you could walk me through it, that would be great. Thanks.

(a) F prime and F double prime nonzero and the same sign?

I assume F is function, F prime is 1st derivative, and F double pime
is 2nd derivative.

F prime is:
>>>Positive if the imaginary tangent line to the curve at that point
is leaning forward, or to the right, as we "go forward or to the
right".
>>>Negative if the imaginary tangent line to the curve at that point
is leaning backward, or to the left, as we "go forward or to the
right".

F double prime is:
>>>Positive if the concavity of the curve at that point is upward, or,
the curve is looking up, is concaved upwards.
>>>Negative if the concavity of the curve at that point is downward,
or, the curve is looking down, is concaved downwards.

Hence, from the picture:
>>>At point B, both F prime anf F double prime are non-zero and positive. 
>>>At point E, both F prime anf F double prime are non-zero and negative.
 
--------------------
(b) At least two of F, F prime, and F double prime equal to zero?

F is zero when the graph touches or intersects the horizontal axis (the x-axis).

F pime is zero when the imaginary tangent line at that point is
horizontal. (Slope of horizontal line is zero.)

F double prime is zero at inflection points, or where the concavity of
the curve changes---from upward to downward, or from downward to
upward.

Hence, from the picture:
>>>At point A, both F and F prime are zero.
>>>At point D, both F prime anf F double prime are zero.

Thank you.