1. ## Anti-derivatives & graphing

I am having trouble with this problem. I asked it on Yahoo Answers, but the responses were not very clear. If someone could explain to me how to approach this problem, that'd be great. Thanks

2. Originally Posted by DanieL34749

I am having trouble with this problem. I asked it on Yahoo Answers, but the responses were not very clear. If someone could explain to me how to approach this problem, that'd be great. Thanks
I might be able to help. In your attachment you are given that

$g(x) = \int_a^x f(t)\, dt$. You are asked about the graph of

$y = \frac{df}{dx}\; = \;\frac{d^2 g}{dx^2}$ from above

The behaviour of y is determine from the behaviour of g(x) and in particular the concavity of g(x). From the picture we see that g(x) goes from concave down to concave up to concave down or

$g'' <0,\;\;\; g''>0,\;\;\; g'' < 0$

so y will go from negative to postive to negative. Which one of your pictures does that?

3. A is the correct answer

4. Originally Posted by nyasha
As why do you say that? It goes contrary to why I just posted.

5. Originally Posted by danny arrigo
As why do you say that? It goes contrary to why I just posted.

After taking a good look at the graph l changed it to "A"

6. Originally Posted by nyasha
After taking a good look at the graph l changed it to "A"
I don't think A is right. Can you give an explanation for your claim?

7. g(x) has a quartic equation and the derivative g'(x) will be a cubic function. This will narrow down your options to A or D. Then from there you look at the end behavior of the polynomial function which will show you that the correct g'(x) is A

8. Originally Posted by nyasha
g(x) has a quartic equation and the derivative g'(x) will be a cubic function. This will narrow down your options to A or D. Then from there you look at the end behavior of the polynomial function which will show you that the correct g'(x) is A
It's not asking about $g'$, it's asking about $f'$ and $f' = g''$.

9. Hey, thanks for the help Danny.

After reading your explanation, I see that A is the one that goes from negative, to positive, to negative. And I understand how the two are tied together too.