Find and classify the local extrema.

01 / Standard#Your turn

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

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

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Solve f′=0.

Hint 2

Examine the derivative signs across −2 and 2.

Worked solution

Factor the derivative and read its signs +,−,+.
$f'(x)=3(x-2)(x+2)$
Evaluate the two stationary values.
$f(-2)=22\text{ (local maximum)},\quad f(2)=-10\text{ (local minimum)}$

Maximum 22 at x=−2; minimum −10 at x=2.

Checks and common pitfalls: These are local extrema; the cubic has no global bounds.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Factor the derivative and read its signs +,−,+.
$f'(x)=3(x-2)(x+2)$
Evaluate the two stationary values.
$f(-2)=22\text{ (local maximum)},\quad f(2)=-10\text{ (local minimum)}$

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

2023 JM02