Sketch the cubic using the preceding derivative information.

01 / Standard#Your turn

f(x)=x^3-12x+6

Official paper · jm02-2023 · 2(a)(iv) · PDF 4

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Mark the maximum, minimum and inflection.

Hint 2

The leading term makes the curve fall left and rise right.

Worked solution

Increase on (−∞,−2) and (2,∞); decrease between them.
$(-2,22),\quad(2,-10),\quad(0,6)$
Draw concave down for x<0 and concave up for x>0, with unbounded cubic tails.
$x\to-\infty:f(x)\to-\infty;\quad x\to\infty:f(x)\to\infty$

The plot marks all required features; the curve continues beyond the viewing window.

Checks and common pitfalls: The inflection point differs from either extremum.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Increase on (−∞,−2) and (2,∞); decrease between them.
$(-2,22),\quad(2,-10),\quad(0,6)$
Draw concave down for x<0 and concave up for x>0, with unbounded cubic tails.
$x\to-\infty:f(x)\to-\infty;\quad x\to\infty:f(x)\to\infty$

Think first. Reveal a hint when the class is ready.

2023 JM02