Two curves can both be rising, yet curve in opposite ways: one bows up like a cup, the other arches over like a dome. This bending is called concavity, and the second derivative f''(x) measures it.

• f''(x) > 0: concave up \smile — the curve holds water; the slope is increasing.
• f''(x) < 0: concave down \frown — the curve spills water; the slope is decreasing.

Concavity is just "the slope of the slope": f''>0 means f' is going up, so the tangents tilt steeper and steeper upward.

Feel the bend

Below is f(x) = x^3 - 3x (bold) with f''(x) = 6x (thin). Slide the tangent: on the left half, where f''<0, the curve is a frown — concave down — and the tangent lies above it; on the right half, where f''>0, it's a smile — concave up — and the tangent lies below.

Inflection points

An inflection point is where the concavity switches — the curve changes from cup to dome or back. There f''(x)=0 (or is undefined) and actually changes sign. For f(x)=x^3-3x:

Just like with extrema, f''=0 is only a candidate — you must check the sign really flips.

The second-derivative test

Concavity gives a quick shortcut for classifying a critical point where f'(c)=0: just check the bend there.

• f''(c) > 0 (concave up, a cup): local minimum.
• f''(c) < 0 (concave down, a dome): local maximum.
• f''(c) = 0: inconclusive — fall back to the first-derivative test.

For f(x)=x^3-3x at x=1: f''(1)=6>0, concave up, so it's a minimum — matching what the sign line told us. Step the diagram to see both critical points classified by their bend.

Watch it on Khan Academy