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?