Concavity and the Second Derivative Test

Concavity is simply which way a graph bends: curving up like a cup, or curving down like an upside-down cup (a cap). All we're really asking is whether the slope is getting bigger or smaller as you move right. The symbols are small pieces: $f''$ (the derivative of the derivative, so it tells you how the slope is changing), $f'' > 0$ (the slope is growing, so the graph bends up) and $f'' < 0$ (the slope is shrinking, so it bends down). An inflection point is just where the bending switches from one to the other. It's the same trick as last lesson, applied one level up.

Last lesson, the sign of $f'$ told us whether $f$ was going up or down. Now apply that same idea to $f'$ itself. If $f'' > 0$, then $f'$ is increasing, so the slopes get steeper as you go: the graph curves upward. If $f'' < 0$, the slopes keep dropping and the graph curves downward.

Concave up means the slope is increasing: the graph lies above its tangent lines, like a cup that holds water. Concave down means the slope is decreasing: the graph lies below its tangents.

Inflection points are where the concavity actually changes. Find where $f'' = 0$ or is undefined, then check that the sign of $f''$ really flips.

A common mistake. It's tempting to think every point with $f'' = 0$ is an inflection point, because that's where you go looking for them. It isn't: $f(x) = x^4$ has $f''(0) = 0$, but $f'' = 12x^2$ is positive on both sides, so the graph is a cup the whole way through and never switches.

The second derivative test is often faster than a sign chart. Think about why it works: if the graph is flat at $c$ and bending up like a cup, the flat spot has to be the bottom of the cup, so it's a minimum. If it's flat and bending down like a cap, you're at the top, so it's a maximum. It fails when $f''(c) = 0$: $x^4$, $-x^4$ and $x^3$ all have $f'(0) = f''(0) = 0$ but behave differently, so in that case go back to the first derivative test.

An inflection point marks diminishing returns: the moment an epidemic's new cases stop accelerating, or where extra ad spending starts buying fewer new customers per dollar.