Hint
A square in h′ can touch zero without changing sign. Use h″ on both sides to investigate concavity.
Worked solution
(a)
The linear inner derivative is 1, so \(\displaystyle h'=3(x-2)^{2}\) and h″=6(x−2).
Answer: \(h^{\prime}=3(x-2)^2\), \(h^{\prime\prime}=6(x-2)\).
(b)
\(\displaystyle h'=0\) only at \(\displaystyle x=2\) and \(\displaystyle h(2)=4\). The derivative is positive on both sides, so the function increases through the point and has no local extremum there. Also h″ is negative before 2 and positive afterwards, proving a change of concavity. Hence (2,4) is a stationary point of inflection. The value h″\(\displaystyle (2)=0\) alone would not prove this.
Answer: \((2,4)\), a stationary point of inflection.
(c)
\(\displaystyle 3(x-2)^{2}\ge 12\) gives \(\displaystyle (x-2)^{2}\ge 4\), so \(\displaystyle x-2\le -2\) or \(\displaystyle x-2\ge 2\). Thus \(\displaystyle x\le 0\) or \(\displaystyle x\ge 4\), including the equality endpoints.
Answer: \(x\le0\) or \(x\ge4\).
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official examination-board mark scheme. Award stated criteria once each, up to the part maximum. Accept valid equivalent methods. Follow-through applies only where explicitly stated; do not award an accuracy criterion for an incorrect final result.
(a) · 2 marks
- 1 mark: Give the correct first derivative.
- 1 mark: Give the correct second derivative.
Alternative methods: Expanding the cubic first is accepted.
Common error: Losing the shift x−2 during differentiation.
(b) · 3 marks
- 1 mark: Find the stationary point (2,4).
- 1 mark: Use the unchanged positive derivative sign or equivalent value comparisons to exclude a local extremum.
- 1 mark: Show the concavity change and classify a stationary point of inflection.
Alternative methods: The shifted-cubic form together with a correct monotonicity and concavity explanation is accepted.
Common error: Calling every point with \(\displaystyle h'=h\)″=0 a point of inflection without further reasoning.
(c) · 2 marks
- 1 mark: Reduce to \(\displaystyle (x-2)^{2}\ge 4\) and select the outside branches.
- 1 mark: Give \(\displaystyle x\le 0\) or \(\displaystyle x\ge 4\) with both endpoints included.
Alternative methods: A correct graph or absolute-value inequality argument is accepted.
Common error: Selecting the interval between the roots or excluding equality.
Total: 7 marks.