Question Think of a ski run: height versus horizontal distance. While you're still skiing downhill, height is decreasing — that's f' negative. But you might be speeding up or slowing down depending on how the slope itself is changing. f' tells increase versus decrease. f'' tells how the slopes are changing — concavity. When f'' > 0 the graph bends like a cup (concave up); when f'' < 0 it bends like a hilltop (concave down).
Consider
Link f' signs to increase/decrease of f Link f'' signs to concavity Identify inflection as a concavity change (not merely f''=0) Narrate motion using f, f', and f'' together
select rect note arrow text clear
Ski height f(x) f' < 0 downhill f'' ? steeper vs gentler f' : up/down f'' : bend inflection: bend flip
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