In examining the graphs of differentiable real functions, it may be useful to state the intervals where the function is convex and the ones where it is concave.

• A function $f$ is said to be convex on an interval if the restriction of $f$ to this interval is a (strictly) convex function; this may be characterized more illustratively by saying that the graph of $f$ is concave upwards or concave up. On such an interval, the tangent line of the graph is constantly turning counterclockwise, i.e., the derivative $f'$ is increasing and thus the second derivative $f''$ is positive. In the picture below, the sine curve is concave up on the interval $(-\pi,\,0)$ .
• The concavity of the function $f$ on an interval correspondingly: On such an interval, the graph of $f$ is concave downwards or concave down, the tangent line turns clockwise, $f'$ decreases, and $f''$ is negative. In the picture below, the sine curve is concave down on the interval $(0,\,\pi)$ .
• The points in which a function changes from concave to convex or vice versa are the inflexion points (or inflection points) of the graph of the function. At an inflexion point, the tangent line crosses the curve, the second derivative vanishes and changes its sign when one passes through the point.

Since the sine function is $2\pi$ -periodic, the sinusoid possesses infinitely many inflexion points. Indeed, $f(x) = \sin x$ ; $f''(x) = -\sin x = 0$ for $x = 0,\,\pm\pi,\,\pm2\pi,\,\dots$ ; $f'''(x) = -\cos x$ , $f'''(n\pi) = -\cos n\pi = (-1)^{n+1} \neq 0$ . Non-nullity of the third derivative at these critical points assures us the existence of those inflexion points.

Remarks

1. For finding the inflexion points of the graph of $f$ it does not suffice to find the roots of the equation $f''(x) = 0$ , since the sign of $f''$ does not necessarily change as one passes such a root. If the second derivative maintains its sign when one of its zeros is passed, we can speak of a plain point (?) of the graph. E.g. the origin is a plain point of the graph of $x\mapsto x^4$ .

2. Recalling that the curvature $\kappa$ for a plane curve $y = f(x)$ is given by $$\kappa(x) \;=\; \frac{f''(x)}{[1+f'(x)^2]^{3/2}},$$ we can say that the inflexion points are the points of the curve where the curvature changes its sign and where the curvature equals zero.

3. If an inflexion point $x = \xi$ satisfies the additional condition $f'(\xi) = 0$ , the point is said to be a stationary inflexion point or a saddle-point, while in the case $f'(\xi) \neq 0$ it is a non-stationary inflexion point.