Moderate -0.3 This is a straightforward application of differentiation to find stationary points. Students need to differentiate y = x + 2cos x to get dy/dx = 1 - 2sin x, solve 1 - 2sin x = 0 for sin x = 1/2 in the given range, then use the second derivative test. While it requires multiple steps, each is routine and the question follows a standard textbook pattern with no novel insight required.
3 The equation of a curve is \(y = x + 2 \cos x\). Find the \(x\)-coordinates of the stationary points of the curve for \(0 \leqslant x \leqslant 2 \pi\), and determine the nature of each of these stationary points.
Carry out appropriate method for determining nature of stationary point
M1
Show that \(x = \frac{1}{6}\pi\) is a maximum with no errors seen
A1
Obtain second answer \(x = \frac{5}{6}\pi\) in range
A1\(\sqrt{}\)
Show this is a minimum point
A1\(\sqrt{}\)
Total: 7
[f.t. is on the incorrect derivative \(1 + 2\sin x\)]
## Question 3:
| Answer/Working | Mark | Guidance |
|---|---|---|
| State correct derivative $1 - 2\sin x$ | B1 | |
| Equate derivative to zero and solve for $x$ | M1 | |
| Obtain answer $x = \frac{1}{6}\pi$ | A1 | |
| Carry out appropriate method for determining nature of stationary point | M1 | |
| Show that $x = \frac{1}{6}\pi$ is a maximum with no errors seen | A1 | |
| Obtain second answer $x = \frac{5}{6}\pi$ in range | A1$\sqrt{}$ | |
| Show this is a minimum point | A1$\sqrt{}$ | Total: 7 |
| | | [f.t. is on the incorrect derivative $1 + 2\sin x$] |
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3 The equation of a curve is $y = x + 2 \cos x$. Find the $x$-coordinates of the stationary points of the curve for $0 \leqslant x \leqslant 2 \pi$, and determine the nature of each of these stationary points.
\hfill \mbox{\textit{CAIE P2 2006 Q3 [7]}}