Find all a for which f is increasing throughout (2,4).

01 / Standard#Your turn

f(x)=ax^2+4x+1

Official paper · jm01-2023 · I.7 · PDF 2

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

For a≥0 the derivative is positive on the interval.

Hint 2

For a<0 its smallest limiting derivative occurs at x=4.

Worked solution

Differentiate.
$f'(x)=2ax+4$
In the negative-a case require 8a+4≥0; equality is allowed because 4 is excluded.
$a\ge-1/2$

D: a≥−1/2.

Checks and common pitfalls: The endpoint case a=−1/2 is still strictly increasing on the open interval.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Differentiate.
$f'(x)=2ax+4$
In the negative-a case require 8a+4≥0; equality is allowed because 4 is excluded.
$a\ge-1/2$

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

2023 JM01