Locate and justify the inflection point.

01 / Standard#Your turn

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

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

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Solve f″(x)=0.

Hint 2

Check that f″ changes sign there.

Worked solution

The second derivative is negative to the left and positive to the right.
$f''(x)=6x-2=0\iff x=\frac13$
Evaluate the original function, not its derivative.
$f\left(\frac13\right)=\frac1{27}-\frac19-\frac13=-\frac{11}{27}$

Inflection point (1/3,−11/27).

Checks and common pitfalls: f″=0 alone does not guarantee an inflection point without a change of concavity.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
The second derivative is negative to the left and positive to the right.
$f''(x)=6x-2=0\iff x=\frac13$
Evaluate the original function, not its derivative.
$f\left(\frac13\right)=\frac1{27}-\frac19-\frac13=-\frac{11}{27}$

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

2026 JM02