Sketch y=V²(x) on the radius domain using the extrema and inflection.

01 / Standard#Your turn

V(x)=\frac\pi3x^2\sqrt{1-x^2},\quad0\le x\le1

Official paper · jm02-2024 · 2(a)(iv) · PDF 4

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Mark (0,0) and (1,0).

Hint 2

The inflection occurs before the maximum.

Worked solution

The graph starts flat at zero, rises to its maximum and falls to zero.
$0<\sqrt{2/5}<\sqrt{2/3}<1$
It is concave up before √(2/5) and concave down afterwards.
$\max y=4\pi^2/243,\quad y(0)=y(1)=0$

The plot shows the full domain, maximum, and inflection.

Checks and common pitfalls: The vertical coordinate is volume squared, not volume.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
The graph starts flat at zero, rises to its maximum and falls to zero.
$0<\sqrt{2/5}<\sqrt{2/3}<1$
It is concave up before √(2/5) and concave down afterwards.
$\max y=4\pi^2/243,\quad y(0)=y(1)=0$

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

2024 JM02