How to find inflection points

Locating inflection points is entirely analogous to locating maxima and minima, but with the second derivative in place of the first: we look for the points where $f''$ vanishes and check that it changes sign.

Property — How to find inflection points

1. Compute $f''(x)$.
2. Find the points where $f''(x)=0$ (or where $f''$ does not exist).
3. Check that $f''$ changes sign around that point. If it does: inflection point. If not: it is not an inflection point (it is a point where the concavity “pauses” but does not change).

The third step is essential: as for extrema, the vanishing of $f''$ is only a necessary condition. If $f''$ does not change sign, the concavity stays the same and the point is not an inflection point.

Example

Let us look for the inflection points of $f(x) = \dfrac{2x+1}{x^2}$. The first derivative $f'(x) = -\dfrac{2}{x^2}-\dfrac{2}{x^3} = -\dfrac{2x+2}{x^3}$ vanishes at $x=-1$ (a stationary point, to be classified by studying the sign of $f'$). The second derivative is $f''(x) = \frac{4}{x^3}+\frac{6}{x^4} = \frac{4x+6}{x^4}, \qquad f''=0 \iff x = -\frac{3}{2}.$ Since $x^4>0$, the sign of $f''$ is that of $4x+6$: for $x < -\tfrac{3}{2}$ we have $f''<0$ (concave), for $-\tfrac{3}{2} < x < 0$ we have $f''>0$ (convex). The concavity changes, so $x=-\dfrac{3}{2}$ is an inflection point.

Links

Topics: Curve sketching
Concepts: Concavity and convexity · Second derivative · Inflection point
Skills: Sketching a function