Convexity and concavity

If the gradient of a curve is increasing, we say it is convex; if the gradient of a curve is decreasing, we say it is concave:

I remember these words by the fact that a concave section of curve looks a bit like a cave.

We have noted before (in the tutorial on finding turning points) that, if $\frac{d y}{d x}$ is increasing, its rate of change must be positive, so

\begin{aligned} convex \\ \Leftrightarrow & \frac{d y}{d x} is increasing \\ \Leftrightarrow & \frac{d}{d x} (\frac{d y}{d x}) \geq 0 \\ \Leftrightarrow & \frac{d^{2} y}{d x^{2}} \geq 0 \end{aligned}

Similarly, for concave sections of curve, we have

\begin{aligned} concave \\ \Leftrightarrow & \frac{d y}{d x} is decreasing \\ \Leftrightarrow & \frac{d}{d x} (\frac{d y}{d x}) \leq 0 \\ \Leftrightarrow & \frac{d^{2} y}{d x^{2}} \leq 0 \end{aligned}

So, in summary:

\begin{aligned} convex & \Leftrightarrow \frac{d^{2} y}{d x^{2}} \geq 0 \\ concave & \Leftrightarrow \frac{d^{2} y}{d x^{2}} \leq 0 \end{aligned}

Points of inflection

A point of inflection is a point where a curve stops being convex and becomes concave, or vice versa. In the diagram below, $P$ is a point of inflection. The curve is concave to the left of $P,$ and convex to the right.

At a point of inflection, the gradient of the tangent is not changing, and therefore

\frac{d}{d x} (\frac{d y}{d x}) = 0

which is better written as

\frac{d^{2} y}{d x^{2}} = 0

A curve is given by

y = - \frac{x^{3}}{3} + 6 x^{2} - 32 x

Find the range of values of $x$ for which the curve is concave.
Sketch the curve, indicating only the coordinates of the point of inflection.

Points of inflection ​

Points of inflection