Points of inflection

Definition 8.8.1   Points of inflection separate arcs concave upwards from arcs concave downwards. They may also be defined as points where

(a) $\frac{d^2 y}{dx^2} = 0$ and $\frac{d^2 y}{dx^2}$ changes sign,

or

(b) $\frac{d^2 x}{dy^2} = 0$ and $\frac{d^2 x}{dy^2}$ changes sign.

Thus, if a curve $y = f(x)$ changes from concave upwards to concave downwards at a point, or the reverse, then such a point is called a point of inflection.

From the discussion of §8.6, it follows at once that where the curve is concave up, $f''(x) = +$, and where the curve is concave down, $f''(x) = -$. In order to change sign it must pass through the value zero^8.8; hence we have:

Lemma 8.8.1   At points of inflection, $f''(x) = 0$.

Solving the equation resulting from Lemma 8.8.1 gives the abscissas of the points of inflection. To determine the direction of curving or direction of bending in the vicinity of a point of inflection, test $f''(x)$ for values of $x$, first a trifle less and then a trifle greater than the abscissa at that point.

If $f''(x)$ changes sign, we have a point of inflection, and the signs obtained determine if the curve is concave upwards or concave downwards in the neighborhood of each point of inflection.

The student should observe that near a point where the curve is concave upwards the curve lies above the tangent, and at a point where the curve is concave downwards the curve lies below the tangent. At a point of inflection the tangent evidently crosses the curve.

Following is a rule for finding points of inflection of the curve whose equation is $y = f(x)$. This rule includes also directions for examining the direction of curvature of the curve in the neighborhood of each point of inflection.

• FIRST STEP. Find $f''(x)$.

• SECOND STEP. Set $f''(x) = 0$, and solve the resulting equation for real roots.

• THIRD STEP. Write $f''(x)$ in factor form.

• FOURTH STEP. Test $f''(x)$ for values of $x$, first a trifle less and then a trifle greater than each root found in the second step. If $f''(x)$ changes sign, we have a point of inflection.

When $f''(x) = +$, the curve is concave upwards^8.9.

When $f''(x) = -$, the curve is concave downwards.