Find the local maximum and minimum values of f.

01 / Standard#Your turn

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

Official paper · jm02-2021 · 2(a)(ii) · PDF 4

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Factor the first derivative to find stationary points.

Hint 2

Read the derivative signs on the three intervals.

Worked solution

The stationary abscissae are 1 and 2.
$f'(x)=6(x-1)(x-2)=0\iff x=1,2$
The signs are positive, negative, positive.
$x<1:f'>0;\quad1<x<2:f'<0;\quad x>2:f'>0$
Evaluate f at the two points and classify the changes.
$f(1)=0\text{ (local maximum)},\quad f(2)=-1\text{ (local minimum)}$

Local maximum 0 at x=1; local minimum −1 at x=2.

Checks and common pitfalls: These local values are not the global bounds of the cubic on the real line.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
The stationary abscissae are 1 and 2.
$f'(x)=6(x-1)(x-2)=0\iff x=1,2$
The signs are positive, negative, positive.
$x<1:f'>0;\quad1<x<2:f'<0;\quad x>2:f'>0$
Evaluate f at the two points and classify the changes.
$f(1)=0\text{ (local maximum)},\quad f(2)=-1\text{ (local minimum)}$

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

2021 JM02