Add 2x+2√y+3z=a to the k=3 system. Find the largest possible a and the solution attaining it.

01 / Standard#Your turn

Official paper · jm02-2024 · 5(c) · PDF 7

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Substitute (x,y,z)=(1−2t,t,t) and require t≥0.

Hint 2

Complete a square in √t.

Worked solution

Reduce the final equation with its domain.
$t\ge0,\quad a=2(1-2t)+2\sqrt t+3t=2-t+2\sqrt t$
The square has minimum zero.
$a=3-(\sqrt t-1)^2\le3,\quad a=3\iff t=1$
Substitute the equality case.
$(x,y,z)=(-1,1,1)$

Maximum a=3, attained only at (−1,1,1).

Checks and common pitfalls: The square root requires y=t≥0.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Reduce the final equation with its domain.
$t\ge0,\quad a=2(1-2t)+2\sqrt t+3t=2-t+2\sqrt t$
The square has minimum zero.
$a=3-(\sqrt t-1)^2\le3,\quad a=3\iff t=1$
Substitute the equality case.
$(x,y,z)=(-1,1,1)$

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

2024 JM02