Hint
The function is continuous on the full bracket. Substitute the lower bracket endpoint.
Worked solution
(a)
\(\displaystyle f(n)=-3<0\) and f(n+1)=59>0. The polynomial is continuous and its derivative \(\displaystyle 3x^{2}+1\) is positive everywhere.
Answer: \(\left(4, 5\right)\)
(b)
Evaluate the midpoint; retain precisely the half whose endpoints still have opposite signs.
Answer: \(\displaystyle \left(4, \frac{9}{2}\right)\)
(c)
Differentiate the polynomial to help investigate uniqueness.
\(49\)
Answer: \(49\)
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) · 4 marks
- 1 mark: Evaluate both endpoint values and establish their opposite signs.
- 1 mark: Use continuity to prove existence in the open interval.
- 1 mark: Use a strictly positive derivative to prove uniqueness.
- 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: A sign change across a pole does not guarantee a zero.
(b) · 2 marks
- 1 mark: Evaluate the midpoint and retain the half with opposite endpoint signs.
- 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: A small function value is not by itself a bound on root error.
(c) · 2 marks
- 1 mark: Differentiate the polynomial to help investigate uniqueness.
- 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: A sign change across a pole does not guarantee a zero.
Total: 8 marks.