Further applications of differentiation — Question 276

ANSWER

a) $(-1,2)$ maximum and $(1,-2)$ minimum. b) $(0,0)$. c) Intercepts $(-\sqrt3,0)$, $(0,0)$, $(\sqrt3,0)$.

Worked solution

a) $f^{\prime}=3x^2-3=0$ at $x=\pm1$. Since $f^{\prime\prime}=6x$, $(-1,2)$ is a maximum and $(1,-2)$ is a minimum.

b) $f^{\prime\prime}$ changes sign at $x=0$ and $f(0)=0$.

c) Solve $x(x^2-3)=0$. Draw a cubic rising from bottom left to top right through these intercepts and turning points.

Exact answers unless rounding is specified; equivalent forms accepted.

a) Solves the stationary equation.

[1 mark]

a) Determines both coordinates.

[1 mark]

a) Correctly classifies both.

[1 mark]

b) Determines (0,0) with a change in concavity.

[1 mark]

c) Labels all intercepts.

[1 mark]

c) Draws the correct cubic shape.

[1 mark]

Practice question aligned to the current QCAA syllabus; review the worked solution and marking criteria.