Find the local maximum and minimum.

01 / Standard#Your turn

f(x)=x^3-3x^2+2

Official paper · jm02-2022 · 2(a)(iii) · PDF 4

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Solve 3x(x−2)=0.

Hint 2

Read the derivative sign changes.

Worked solution

Derivative signs across 0,2 are +,−,+.
$f'(x)=3x(x-2)$
Evaluate the corresponding function values.
$f(0)=2\text{ (local maximum)},\quad f(2)=-2\text{ (local minimum)}$

Local maximum 2 at x=0; local minimum −2 at x=2.

Checks and common pitfalls: These are local, not global, extrema of the cubic.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Derivative signs across 0,2 are +,−,+.
$f'(x)=3x(x-2)$
Evaluate the corresponding function values.
$f(0)=2\text{ (local maximum)},\quad f(2)=-2\text{ (local minimum)}$

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

2022 JM02