The simulation at

http://phet.colorado.edu/sims/calculus-grapher/calculus-grapher_en.html

allows you to construct graphs and see how the corresponding derivative and integral functions behave.

Use the free-form tool (you can find it easily enough; names of the tools pop up on mouse-over).  Check both the integral and derivative boxes.  Construct a graph that is concave up.  Describe the resulting derivative and integral functions.

Now see if you can construct a graph whose derivative is concave up.  How did you manage this?  What does your graph look like?  What does the integral function look like?

Now try to construct a graph whose integral is concave down.  How did you manage this?  What does your graph look like?  What does the derivative function look like?