Hint
Use the stated units and precision. Solve the zero-gradient equation, then substitute into the original cubic for any requested ordinate.
Worked solution
(a)
Solve \(12 x^{2} - 108 = 0\). The second derivative is \(24 x\), negative at the smaller root and positive at the larger. Substitute in f for both y-coordinates.
Answer: (-3,221) maximum; (3,-211) minimum.
(b)
Solve the zero-gradient equation, then substitute into the original cubic for any requested ordinate.
\(x^2=9\) gives both signed roots; the larger is 3.
Answer: \(3\)
(c)
Solve the zero-gradient equation, then substitute into the original cubic for any requested ordinate.
At \(\displaystyle x=-3, f=221\) and f″=-72<0.
Answer: \(221\)
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit complete valid alternative methods and consequential reasoning. An unexplained result does not earn a method mark.
(a) · 3 marks
- 1 mark: Solve \(\displaystyle f'=0\) retaining both roots.
- 1 mark: Substitute into f and classify each point using a valid derivative test.
- 1 mark: Give the complete requested result, using the required units, precision and boundary inclusion.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Keep intermediate values exact until the final step.
(b) · 2 marks
- 1 mark: Solve the zero-gradient equation, then substitute into the original cubic for any requested ordinate.
- 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: Check the answer against the original conditions.
(c) · 2 marks
- 1 mark: Solve the zero-gradient equation, then substitute into the original cubic for any requested ordinate.
- 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: Keep intermediate values exact until the final step.
Total: 7 marks.