Inflection point

For the graph of a function f of differentiability class C2 (its first derivative f', and its second derivative f'', exist and are continuous), the condition f'' = 0 can also be used to find an inflection point since a point of f'' = 0 must be passed to change f'' from a positive value (concave upward) to a negative value (concave downward) or vice versa as f'' is continuous; an inflection point of the curve is where f'' = 0 and changes its sign at the point (from positive to negative or from negative to positive).

[2][3] For example, the graph of the differentiable function has an inflection point at (x, f(x)) if and only if its first derivative f' has an isolated extremum at x.

For a smooth curve given by parametric equations, a point is an inflection point if its signed curvature changes from plus to minus or from minus to plus, i.e., changes sign.

In this case, one also needs the lowest-order (above the second) non-zero derivative to be of odd order (third, fifth, etc.).

In the preceding assertions, it is assumed that f has some higher-order non-zero derivative at x, which is not necessarily the case.

If it is the case, the condition that the first nonzero derivative has an odd order implies that the sign of f'(x) is the same on either side of x in a neighborhood of x.

The tangent at the origin is the line y = ax, which cuts the graph at this point.

is concave for negative x and convex for positive x, but it has no points of inflection because 0 is not in the domain of the function.

Plot of y = x 3 with an inflection point at (0,0), which is also a stationary point .
The roots , stationary points , inflection point and concavity of a cubic polynomial x 3 − 6 x 2 + 9 x − 4 (solid black curve) and its first (dashed red) and second (dotted orange) derivatives .
Plot of f ( x ) = sin(2 x ) from − π /4 to 5 π /4; the second derivative is f″ ( x ) = –4sin(2 x ) , and its sign is thus the opposite of the sign of f . Tangent is blue where the curve is convex (above its own tangent ), green where concave (below its tangent), and red at inflection points: 0, π /2 and π
y = x 4 – x has a 2nd derivative of zero at point (0,0), but it is not an inflection point because the fourth derivative is the first higher order non-zero derivative (the third derivative is zero as well).