Hint
Give coordinates, not just stationary abscissae. Set the factored first derivative equal to zero.
Worked solution
(a)
\(12 \left(x - 2\right)^{2} - 108\) vanishes at \(-1\) and \(5\). Substitute each into f.
Answer: \(\left( -1, \ 228\right)\) and \(\left( 5, \ -204\right)\)
(b)
The second derivative is \(24 \left(x - 2\right)\). Its values are \(-72\) and \(72\), respectively.
Answer: The first is a local maximum; the second is a local minimum.
(c)
Evaluate f at h−d; the second derivative is negative.
\(228\)
Answer: \(228\)
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit complete valid alternatives and consequential reasoning; an unexplained final answer does not earn a method mark.
(a) · 2 marks
- 1 mark: Solve \(\displaystyle y'=0\) for both abscissae and evaluate both ordinates.
- 1 mark: Give all results requested in this part, with the correct signs, exact values, units, branch and boundary inclusion where applicable.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: The derivative value is zero, not the point height.
(b) · 2 marks
- 1 mark: Use a derivative-sign change or nonzero second derivative at each stationary point.
- 1 mark: Give all results requested in this part, with the correct signs, exact values, units, branch and boundary inclusion where applicable.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Local extrema need not be global extrema.
(c) · 2 marks
- 1 mark: Evaluate f at h−d; the second derivative is negative.
- 1 mark: Give the correct requested result, retaining signs, units and any domain restrictions.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: The derivative value is zero, not the point height.
Total: 6 marks.