Find and classify the local extrema.

01 / Standard#Your turn

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

Official paper · jm02-2026 · 2(a)(ii) · PDF 4

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Solve f′(x)=0.

Hint 2

Check the derivative sign on each side of both roots.

Worked solution

Locate the stationary points.
$f'(x)=(3x+1)(x-1)=0\iff x=-\frac13,1$
The derivative signs are +,−,+, hence a maximum then a minimum.
$f\left(-\frac13\right)=\frac5{27},\quad f(1)=-1$

Local maximum 5/27 at x=−1/3; local minimum −1 at x=1.

Checks and common pitfalls: A stationary x-coordinate is not itself the extremum value.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Locate the stationary points.
$f'(x)=(3x+1)(x-1)=0\iff x=-\frac13,1$
The derivative signs are +,−,+, hence a maximum then a minimum.
$f\left(-\frac13\right)=\frac5{27},\quad f(1)=-1$

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

2026 JM02