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

Think of f' as a report on f's slope and f'' as a report on f''s slope's slope. Where f has a relative maximum or minimum, its tangent line is flat, so f' must equal zero there and cross from one sign to the other — positive to negative at a max, negative to positive at a min. Where f is concave up, its slope is increasing, so f' is increasing and f'' is positive; where f is concave down, f' is decreasing and f'' is negative. An inflection point on f, where concavity flips, shows up as a relative max or min on f' and a sign change on f''.

Stack the three graphs vertically and read straight down. A zero crossing on the f' graph lines up with a peak or valley on the f graph above it. A peak or valley on the f' graph lines up with a zero crossing on the f'' graph below it, and that same x-value is an inflection point on f. Sign changes are the real signal: f' tells you where f increases or decreases, f'' tells you where f curves up or down, and together they tell you the full shape, something no single graph gives alone.

The catch: a peak on f' is not a peak on f. If f' has a relative maximum but stays positive there, f is still climbing, just as fast as it will ever climb at that instant. f only turns where f' itself crosses zero. This distinction, between where a function peaks and where its rate of change peaks, is what separates reading f' correctly from mixing it up with f.