Owl: 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).
Inflection is where concavity switches. And no — f'' = 0 is not automatically an inflection; you need an actual change of concavity.
Here's a playful question: can you be going downhill (f' < 0) while still accelerating in speed along the trail because the slope is getting steeper?