Two functions, h'(x) and h''(x), are graphed below.

(a)    Which graph is which and why?
(b)    What will h'''(x) look like for x > 0? Draw one possible graph.
(c)    What will h(x) look like for x > 0? Draw one possible graph.

Difficulty Rating: 

(a)    The red graph is h''(x), and the blue is h'(x). Notice that the red graph hits the x-axis (has a value of 0) whenever the blue has a relative maximum or minimum. Furthermore, the red graph is negative (below the x-axis) whenever the blue graph is decreasing but positive when the blue graph is increasing. Both of these are good justifications.

(b)    Using the same reasoning as in part (a), this is a reasonably good graph of h'''(x).

(c)    For the same reasons you used in part (a), the following is a reasonably good graph of the original function, h(x).

Please excuse my lack of artistic ability; I know these graphs look as if I drew them with my feet.