Connecting a Function, Its First Derivative, and Its Second Derivative

Each derivative is a report on the behavior of the function before it. f' reports the slope of f: positive means f is increasing, negative means f is decreasing, and a zero crossing marks a peak or valley of f. f'' reports the slope of f', which is the concavity of f: positive means f curves upward like a cup, negative means f curves downward like a dome, and a zero crossing where the sign actually changes marks an inflection point. Three stacked graphs can describe one curve because they're just measuring different things about the same motion — position, velocity-like rate of change, and the rate of change of that rate.

To sort three unlabeled graphs, pick one candidate for f and check it against the others. Where your candidate f is rising, the candidate f' should sit above the x-axis; where f is falling, f' should sit below it. Where f' is rising, f'' should be above the axis; where f' is falling, f'' should be below it. If every region matches, you've found the right stack. This same logic is what lets you move from a formula for f' straight to full concavity and extremum information about f without ever writing down f itself, which is exactly the skill the rest of this unit leans on when hunting for absolute max and min values.