Find the inflection point of y=f(x).

01 / Standard#Your turn

f(x)=2x^3-9x^2+12x-5

Official paper · jm02-2021 · 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 concavity changes there.

Worked solution

The second derivative vanishes at 3/2.
$12x-18=0\implies x=\tfrac32$
It changes from negative to positive; evaluate the ordinate.
$x<\tfrac32:f''<0;\quad x>\tfrac32:f''>0;\quad f(\tfrac32)=-\tfrac12$

Inflection point (3/2,−1/2).

Checks and common pitfalls: An inflection requires a concavity change; f″=0 alone is insufficient.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
The second derivative vanishes at 3/2.
$12x-18=0\implies x=\tfrac32$
It changes from negative to positive; evaluate the ordinate.
$x<\tfrac32:f''<0;\quad x>\tfrac32:f''>0;\quad f(\tfrac32)=-\tfrac12$

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

2021 JM02