Higher Derivatives and Concavity

The Derivative of the Derivative

Differentiating a derivative gives the second derivative $f''(x)$ — the rate of change of the rate of change. If $f$ is position, $f'$ is velocity and $f''$ is acceleration. Higher derivatives ($f'''$, and so on) continue the pattern, each measuring the change of the one before.

The second derivative reveals concavity — how a curve bends:

• $f''(x) > 0$ — the curve is concave up (cup-shaped, holds water); the slope is increasing.
• $f''(x) < 0$ — the curve is concave down (cap-shaped, spills water); the slope is decreasing.

A point where concavity changes sign is an inflection point — the curve switches from bending one way to the other. At an inflection point $f''$ is zero (or undefined), but the reverse is not guaranteed: $f'' = 0$ is necessary, not sufficient, for an inflection (concavity must actually change).

Concavity gives the second-derivative test for classifying critical points (where $f'(x) = 0$):

• $f''(x) > 0$ at a critical point $\Rightarrow$ local minimum (bottom of a cup).
• $f''(x) < 0$ at a critical point $\Rightarrow$ local maximum (top of a cap).
• $f''(x) = 0$ $\Rightarrow$ the test is inconclusive; fall back on the first-derivative sign test.

Together, the first derivative (increasing/decreasing, critical points) and the second (concavity, inflections) determine a curve's full shape, which is the heart of curve sketching and of optimization in the next lesson.

Common pitfall: assuming that $f''(x) = 0$ guarantees an inflection point. A zero second derivative is only a candidate — an inflection requires the concavity to actually change sign across the point. For example $f(x) = x^4$ has $f''(0) = 0$, yet it is concave up on both sides, so $x = 0$ is not an inflection point. Always check that concavity switches.