Inflection points and concavity

A point where the direction in which a graph bends reverses is an inflection point¹. We find it for $f(x) = x^3 - 6x^2 + 9x$.

The second derivative

The direction of bending is decided by the second derivative. Differentiating $f'(x) = 3x^2 - 12x + 9$ once more gives $f''(x) = 6x - 12$.

We have $f'' = 0$ at $x = 2$, and $f(2) = 8 - 24 + 18 = 2$, so the inflection point is $(2, 2)$.

$f''$ measures how the slope changes

$f''$ describes whether $f'$ is rising or falling. Where $f'' < 0$ the slope of the tangent decreases, and where $f'' > 0$ it increases. The inflection point is where that reverses, and here it is the place where the slope of the tangent is least. Indeed $f'(2) = -3$ is the minimum of $f'$.

The center of point symmetry

A cubic has point symmetry about its inflection point. The midpoint of the local maximum $(1, 4)$ and the local minimum $(3, 0)$ is $(2, 2)$, which is the inflection point. Rotating the graph of a cubic by $180^\circ$ about that point carries it onto itself.

Watch for the change of sign

A point where $f'' = 0$ is only a candidate. For $f(x) = x^4$ we have $f''(0) = 0$, but the sign of $f''$ does not change on either side, so it is not an inflection point. As with extrema, the change of sign has to be checked.

How many inflection points are possible

A cubic has only one switch of concavity because its $f''$ is a linear expression.

The tangent at the inflection point

From $f'(2) = -3$ it is $y = -3(x - 2) + 2 = -3x + 8$. Since the concavity turns from down to up there, this tangent crosses the curve at the inflection point.

Correspondence with extrema

Extrema are decided by the sign of $f'$, concavity by the sign of $f''$.

The cubic on the graph is $y = x^3 - 6x^2 + 9x$, the line is $y = f''(x) = 6x - 12$, and the large dot is the inflection point $(2, 2)$.

1. Inflection point, Wikipedia