Standard +0.3 This is a straightforward stationary points question requiring differentiation of a simple trigonometric function (cos 2x), solving a basic trigonometric equation (sin 2x = 1/2), and using the second derivative test. While it involves multiple steps, each is routine for P3 level with no novel problem-solving required, making it slightly easier than average.
3 The equation of a curve is \(y = x + \cos 2 x\). Find the \(x\)-coordinates of the stationary points of the curve for which \(0 \leqslant x \leqslant \pi\), and determine the nature of each of these stationary points.
Carry out an appropriate method for determining the nature of a stationary point
M1
Show that \(x = \frac{1}{12}\pi\) is a maximum with no errors seen
A1
Obtain second answer \(x = \frac{7}{12}\pi\) in range
A1√
Show this is a minimum point
A1
Total: [7]
State correct derivative $1 - 2\sin 2x$ | B1 |
Equate derivative to zero and solve for $x$ | M1 |
Obtain answer $x = \frac{1}{12}\pi$ | A1 |
Carry out an appropriate method for determining the nature of a stationary point | M1 |
Show that $x = \frac{1}{12}\pi$ is a maximum with no errors seen | A1 |
Obtain second answer $x = \frac{7}{12}\pi$ in range | A1√ |
Show this is a minimum point | A1 |
**Total: [7]**
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3 The equation of a curve is $y = x + \cos 2 x$. Find the $x$-coordinates of the stationary points of the curve for which $0 \leqslant x \leqslant \pi$, and determine the nature of each of these stationary points.
\hfill \mbox{\textit{CAIE P3 2005 Q3 [7]}}