Question

If f(x), f'(x), f"(x) are negative what does that tell us about f(x)

Answer 1

f'(x) is negative, I know this tells me f(x) is decreasing
f"(x) is negative, I know this tells me f(x) is concave down
not sure what f(x) is negative what that tells me, does that mean f(x) are in Quadrants III or IV

Answer 2

Yeah, that sounds good to me.

Answer 3

I have to pick the function that fits that description. I picked graph B. Sorry I had to put is as an attachment but my sketches will not do justice

Answer 4

looks like B to me

Answer 5

Yeah that's fine. B is definitely correct, but let's take this a step further so that you have the confidence to understand why all the other ones are wrong.

Which ones are you unsure of as to why they are wrong? Perhaps E or C? For what of the following holds true for these and what doesn't in terms of f, f' and f''? For instance, we know the graph of A is negative, but the derivative is not for all values.

Answer 6

for all the reasons you posted

Answer 7

I don't have the confidence with these problems because I am still learning to read graphs and their first and second derivatives.

Answer 8

That's fine, I'm just saying I'm here to help you.

Answer 9

Not A because portion of graph is increasing
Not C because portion of graph is increasing
Not D because of Quadrant I and II
No E well I don't know why I would eliminate it
Thanks I appreciate all and any help

Answer 10

E just looks wrong but I can't offer any other explanation

Answer 11

heard of `inflection point` right ?

Answer 12

yes but pt of inflection only happens when function changes concavity

Answer 13

Concavity is the second derivative; essentially the "change of the change" So we can see that the graph is decreasing. That's it's change.

Now if we look at B we see it's actually decreasing more and more in the downwards direction.

However if we look at E we see that it decreases, but it decreases less and less as you move to the right! That means it must have a positive concavity there!

Answer 14

guess I am use to seeing the concavity happening like this rather than on its side.

Answer 15

Concavity happens on other graphs than y=x^3. You should completely forget about this because concavity has really no special connection to this graph or any other graph, it's a property of ALL functions you can take the second derivative of.



If you pick any point on this graph notice that the point to the right of it is lower. This is decreasing, f'(x) < 0. Now the second derivative describes how it's decreasing. In a sense, you can think of it as which direction does the graph accelerate?

On the left part of the graph you can pick any point and notice that the neighbor on its left is closer in height than its neighbor on its right. This is because it has a downwards (negative) concavity.