Find the minimum of ab under the positive-variable condition.

01 / Standard#Your turn

5/a+4/b=3,\quad a,b>0

Official paper · jm01-2021 · I.15 · PDF 3

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Let u=5/a and v=4/b; their sum is fixed.

Hint 2

Maximize uv to minimize ab=20/(uv).

Worked solution

Apply AM–GM to the two positive new variables.
$u+v=3\Rightarrow uv\le(3/2)^2=9/4$
Invert the positive bound and verify equality.
$ab=20/(uv)\ge80/9;\quad a=10/3,\ b=8/3$

C: 80/9.

Checks and common pitfalls: Equality requires 5/a=4/b, not a=b.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Apply AM–GM to the two positive new variables.
$u+v=3\Rightarrow uv\le(3/2)^2=9/4$
Invert the positive bound and verify equality.
$ab=20/(uv)\ge80/9;\quad a=10/3,\ b=8/3$

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

2021 JM01