given the graph of $~f'(x)~$ find the concavity of $~f(x)~$

Question

I was doing a calculus problem and I require some clarification/help. So the problem goes as follows:

Given the graph of $~f'(x)~$, find the concavities of $~f(x)~$.

The answer I found was that $~f(x)~$ is concave up from $~(-6,4)\cup (11,∞)~$ and concave down from $~(-∞, -6) \cup (4,11)~$.

However, the answer key said $~f(x)~$ is concave up from $~(-6,1) \cup (1,4) \cup (11,∞)~$ and concave down from $~(-∞, -6) \cup (4,11)~$.

Is there a reason for this?

I think it might have something to do with the removable discontinuity at $~(1,6)~$, but I'm not sure why.

Picture of graph of $~f'(x)~$ is

Answer 1

f is concave up when f$^{\prime\prime}$ > 0. f is increasing when f$^{\prime}$ > 0. Thus, if f$^{\prime}$ is increasing, then f$^{\prime\prime}$ > 0. So you just need to check where f$^{\prime}$ is increasing to know when f is concave up.

Answer 2

At $x=1$, the graph shows that $f’$ is undefined. Therefore, $f’’$ is undefined at $x=1$ too. So we cannot conclude anything about concavity at that point.