Standard +0.3 This is a straightforward application of finding where a cubic function is increasing by setting f'(x) ≥ 0, solving a quadratic inequality, and identifying the critical value. The question requires standard differentiation and completing the square or using the quadratic formula, but involves no novel insight—slightly easier than average due to its routine nature.
The function \(f\) is such that \(f(x) = x^3 - 3x^2 - 9x + 2\) for \(x > n\), where \(n\) is an integer. It is given that \(f\) is an increasing function. Find the least possible value of \(n\). [4]
Attempt to solve f′(x)=0 or f'(x)>0 or f'(x)(cid:46)0 soi
(3)(x−3)(x+1) or 3,−1 seen or 3 only seen
Answer
Marks
Least possible value of n is 3. Accept n = 3. Accept n(cid:46)3
B1
M1
A1
Answer
Marks
Guidance
A1
[4]
With or without
equality/inequality signs
Must be in terms of n
Question 4:
4 | f′(x)=3x2 −6x−9 soi
Attempt to solve f′(x)=0 or f'(x)>0 or f'(x)(cid:46)0 soi
(3)(x−3)(x+1) or 3,−1 seen or 3 only seen
Least possible value of n is 3. Accept n = 3. Accept n(cid:46)3 | B1
M1
A1
A1 | [4] | With or without
equality/inequality signs
Must be in terms of n
The function $f$ is such that $f(x) = x^3 - 3x^2 - 9x + 2$ for $x > n$, where $n$ is an integer. It is given that $f$ is an increasing function. Find the least possible value of $n$. [4]
\hfill \mbox{\textit{CAIE P1 2016 Q4 [4]}}