Moderate -0.8 This is a straightforward application of differentiation to determine monotonicity. Students need to find f'(x) = 3x² + 4x - 4, factorize or use the quadratic formula to find critical points, then check the sign of f'(x) on the given domain x ≥ -2. While it requires completing the discriminant and sign analysis, it's a standard textbook exercise with no novel insight required, making it easier than average.
2 The function f is defined by \(\mathrm { f } ( x ) = x ^ { 3 } + 2 x ^ { 2 } - 4 x + 7\) for \(x \geqslant - 2\). Determine, showing all necessary working, whether f is an increasing function, a decreasing function or neither.
Factors or critical values, or sub any 2 values \((x \neq -2)\) into \(f'(x)\)
M1
Expect \((x+2)(3x-2)\) or \(-2, \frac{2}{3}\) or any 2 subs (excluding \(x=-2\))
For \(-2 < x < \frac{2}{3}\), \(f'(x) < 0\); for \(x > \frac{2}{3}\), \(f'(x) > 0\)
M1
Or at least 1 specific value \((\neq -2)\) in each interval giving opposite signs. Or \(f'(\frac{2}{3})=0\) and \(f''(\frac{2}{3}) \neq 0\)
Neither www
A1
Must have 'Neither'
ALT 1: At least 3 values of \(f(x)\)
M1
e.g. \(f(0)=7\), \(f(1)=6\), \(f(2)=15\)
At least 3 correct values of \(f(x)\)
A1
At least 3 correct values of \(f(x)\) spanning \(x = \frac{2}{3}\)
A1
Shows a decreasing then increasing pattern. Neither www
A1
Or similar wording. Must have 'Neither'
ALT 2: \(f'(x) = 3x^2+4x-4 = 3(x+\frac{2}{3})^2 - \frac{16}{3}\)
B1B1
Do not condone sign errors
\(f'(x) \geq -\frac{16}{3}\)
M1
\(f'(x) < 0\) for some values and \(> 0\) for other values. Neither www
A1
Or similar wording. Must have 'Neither'
## Question 2:
| $f'(x) = 3x^2 + 4x - 4$ | B1 | |
|---|---|---|
| Factors or critical values, or sub any 2 values $(x \neq -2)$ into $f'(x)$ | M1 | Expect $(x+2)(3x-2)$ or $-2, \frac{2}{3}$ or any 2 subs (excluding $x=-2$) |
| For $-2 < x < \frac{2}{3}$, $f'(x) < 0$; for $x > \frac{2}{3}$, $f'(x) > 0$ | M1 | Or at least 1 specific value $(\neq -2)$ in each interval giving opposite signs. Or $f'(\frac{2}{3})=0$ and $f''(\frac{2}{3}) \neq 0$ |
| Neither www | A1 | Must have 'Neither' |
| ALT 1: At least 3 values of $f(x)$ | M1 | e.g. $f(0)=7$, $f(1)=6$, $f(2)=15$ |
| At least 3 correct values of $f(x)$ | A1 | |
| At least 3 correct values of $f(x)$ spanning $x = \frac{2}{3}$ | A1 | |
| Shows a decreasing then increasing pattern. Neither www | A1 | Or similar wording. Must have 'Neither' |
| ALT 2: $f'(x) = 3x^2+4x-4 = 3(x+\frac{2}{3})^2 - \frac{16}{3}$ | B1B1 | Do not condone sign errors |
| $f'(x) \geq -\frac{16}{3}$ | M1 | |
| $f'(x) < 0$ for some values and $> 0$ for other values. Neither www | A1 | Or similar wording. Must have 'Neither' |
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2 The function f is defined by $\mathrm { f } ( x ) = x ^ { 3 } + 2 x ^ { 2 } - 4 x + 7$ for $x \geqslant - 2$. Determine, showing all necessary working, whether f is an increasing function, a decreasing function or neither.\\
\hfill \mbox{\textit{CAIE P1 2018 Q2 [4]}}