Sorry for the long post. Here's the question:
The graph below is that of the derivative of the function $f$ whose domain is the set of all real numbers and is continuous everywhere. Determine the intervals where the graph of $f$ is concave down. Choose all that apply.
Here's the graph. I'm not sure what the function is exactly, sorry!
My choices, and I'm allowed to select multiple, are:
- A) $(-\infty, -2)$
- B) $(-\infty, -3)$
- C) $(-3, 4)$
- D) $(0, \infty)$
- E) $(-3, -1)$
- F) $(1, 4)$
- G) $(2, \infty)$
- H) $(4, \infty)$
After discussing this on r/HomeworkHelp on Reddit, a couple users helped me further understand what I needed to do (since this topic is a little foggy for me), and after carefully working through it, I concluded that my answers are between $(-\infty, -2)$ and between $(2, \infty)$, so it would be $A, B, G, H$, but another user commented and said the second concave down interval is $(0, \infty)$, so the answer would also include $D, F$. I'm not sure which one's correct…
Here's how I got my original answer, the one where the second interval starts at $(2, \infty)$:
The concavity of the intervals on the graph would be down-up-down; these intervals, respectively, being $(-\infty, -2)$, $(-2, 2)$, and $(2, \infty)$, and since it only asks for intervals that concave down, I'm only looking for anything in between (and including) the first and third interval that I mentioned. If I've made a mistake in my thought process or if I'm simply not understanding something, I'd gladly appreciate any tips! Thank you! 🙂
Best Answer
The function is concave down when the second derivative is negative. Since you're given the derivative, you want the intervals where the derivative of the derivative is negative, i.e. the derivative is decreasing. This is clearly the case for the intervals $(-\infty,-2)$ and $(2,\infty)$, and not the case for $[-2,2]$.
Now, clearly these intervals correspond to $A$ and $G$ (resp.). Since $B\subset A$ and $H\subset G$, those also seem to be valid responses. Thus, to my best interpretation of the problem statement,
$$\{A,B,G,H\}$$
is the most correct set of choices.
(The comment that suggested $(0,\infty)$ was likely reading that graph as the function itself, and missed the part of the instructions where it is stated to be the derivative of the function in question)