What’s the graph look like?

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Two young mathematicians discuss how to sketch the graphs of functions.

Check out this dialogue between two calculus students (based on a true story):

Devyn: Riley, I’ve been thinking about the derivative.
Riley: It’s all about change. It’s some “change-detector” tool for math.
Devyn: I know! What’s crazy is that you can use it as a tool for sniffing out dirt on functions.
Riley: First $f'$ tells us increasing or decreasing.
Devyn: Then $f''$ tells us concavity.
Riley: From just that we know all local maxes and mins.
Devyn: And if we use limits, we can find any asymptotes!
Riley: You know, I’d like to make up a procedure based on all these facts, that would tell me what the graph of any function would look like.
Devyn: Me too! Let’s get to work!

On some interval, we know that $f'(x)$ is positive and $f''(x)$ is positive. Which of the following is the best option for the shape of the graph on that interval?

On some interval, we know that $f'(x)$ is negative and $f''(x)$ is positive. Which of the following is the best option for the shape of the graph on that interval?