Determining Concavity of Functions over Their Domains

Two roads can both climb uphill, but one feels smooth because it curves gently upward and the other feels sickening because it curves downward like the top of a hill. That's concavity: the shape of the bend, separate from whether you're going up or down. A function is concave up where its graph bends like a cup, holding water. It's concave down where it bends like a dome, spilling water off.

The second derivative gives you this shape directly. Where f''(x) > 0, the slope f'(x) is increasing, so the curve bends upward: concave up. Where f''(x) < 0, the slope is decreasing, so the curve bends downward: concave down. To find concavity over a domain, take f'', set it equal to zero, solve for the candidate points, then test the sign of f'' on each interval between them. Those sign-change points are called inflection points, and they mark where the bend flips direction.

This sits right next to the increasing/decreasing work from f', but the two ideas answer different questions. f' tells you the direction of travel; f'' tells you the shape of the road. A function can be decreasing and concave up at once, like rolling into the bottom of a valley before leveling off, or increasing and concave down, like a car losing speed as it climbs a hill. Keeping these separate is what lets you sketch a full, accurate picture of a curve's behavior using only its derivatives.