Find and interpret the second derivative, determine concavity and points of inflection, and apply the second derivative test

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.

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.