The second derivative and concavity: WACE Year 12 Mathematics Methods Unit 3

What this dot point is asking

SCSA Unit 3 introduces the second derivative as the derivative of the derivative. This dot point asks you to compute it, read concavity from its sign, locate points of inflection, and use it to classify stationary points, all examinable in both sections of the WACE written examination.

What the second derivative measures

The first derivative $f'(x)$ is the rate of change of $f$. The second derivative $f''(x)$ is the rate of change of $f'$, so it measures how the gradient is changing. In kinematics, if $x(t)$ is displacement then $x'(t)$ is velocity and $x''(t)$ is acceleration.

A common requirement is that $f''(a)=0$ alone does not guarantee an inflection: the sign of $f''$ must actually change there. Confirming the sign change, with a brief sign table or a sentence stating the signs on each side, is the part SCSA markers look for and the part most often omitted.

Points of inflection

To find points of inflection, solve $f''(x)=0$, then confirm a sign change of $f''$ on either side. The function $f(x)=x^{4}$ illustrates the trap: $f''(x)=12x^{2}$ is zero at $x=0$ but never negative, so the concavity does not reverse and there is no inflection.

The second derivative test

Once a stationary point $x=a$ is found from $f'(a)=0$, the second derivative test classifies it quickly.

Acceleration and the second derivative

In kinematics the second derivative has a direct physical meaning. If $x(t)$ is displacement, then $x'(t)=v(t)$ is velocity and $x''(t)=a(t)$ is acceleration. Concave-up displacement ($x''>0$) means the velocity is increasing, and concave-down displacement ($x''<0$) means it is decreasing. A point of inflection of the displacement graph is exactly where the acceleration is zero and changing sign, which is the instant the motion switches between speeding up and slowing down. This connects the abstract idea of concavity to a quantity students can feel.

Reading concavity from a sketch

When you sketch a curve, concavity tells you how to draw it between the key points. A concave-up section curves like a valley with the gradient steadily increasing; a concave-down section curves like a hill with the gradient steadily decreasing. At a point of inflection the tangent crosses the curve as the bend reverses. Getting the concavity right is a marked feature of a curve sketch, separate from plotting the correct stationary points.