Standard +0.8 This requires applying the product rule to a composite function (cos³x · sin^(1/2)x), simplifying the derivative algebraically, setting it to zero, and solving a non-trivial trigonometric equation numerically. The algebraic manipulation and equation-solving go beyond routine differentiation exercises, though the overall structure is standard for stationary point problems.
4 The equation of a curve is \(y = \cos ^ { 3 } x \sqrt { \sin x }\). It is given that the curve has one stationary point in the interval \(0 < x < \frac { 1 } { 2 } \pi\).
Find the \(x\)-coordinate of this stationary point, giving your answer correct to 3 significant figures.
Use correct product rule and chain rule to differentiate either \(\cos^3 x\) or \(\sqrt{\sin x}\)
M1
e.g. two terms with one part of \(\frac{dy}{dx} = p\cos^2 x \sin x\sqrt{\sin x} + q\frac{\cos^3 x \cos x}{\sqrt{\sin x}}\)
Obtain correct derivative in any form e.g. \(\frac{dy}{dx} = -3\cos^2 x \sin x\sqrt{\sin x} + \frac{\cos^3 x \cos x}{2\sqrt{\sin x}}\)
A1 A1
A1 for each correct term substituted in the complete derivative
Equate derivative to zero and obtain horizontal equation with positive integer powers of \(\sin x\) and/or \(\cos x\) from equation including \(\sqrt{\sin x}\) or \(\frac{1}{\sqrt{\sin x}}\) using sensible algebra
M1
e.g. \(-3\cos^2 x \sin^2 x + \frac{1}{2}\cos^4 x = 0\)
Use correct formula(s) to express their equation/derivative in terms of one trigonometric function
M1
Can be awarded before the previous M1. May involve more than one trigonometric term
Obtain \(7\cos^2 x = 6\), \(7\sin^2 x = 1\), or \(6\tan^2 x = 1\), or equivalent, and obtain answer \(x = 0.388\)
A1
CAO. Question asks for 3 sf. Ignore additional answers outside \(\left(0, \frac{\pi}{2}\right)\). \(22.2°\) is A0
Total: 6 marks
## Question 4:
| Use correct product rule and chain rule to differentiate either $\cos^3 x$ or $\sqrt{\sin x}$ | M1 | e.g. two terms with one part of $\frac{dy}{dx} = p\cos^2 x \sin x\sqrt{\sin x} + q\frac{\cos^3 x \cos x}{\sqrt{\sin x}}$ |
|---|---|---|
| Obtain correct derivative in any form e.g. $\frac{dy}{dx} = -3\cos^2 x \sin x\sqrt{\sin x} + \frac{\cos^3 x \cos x}{2\sqrt{\sin x}}$ | A1 A1 | A1 for each correct term substituted in the complete derivative |
| Equate derivative to zero and obtain horizontal equation with positive integer powers of $\sin x$ and/or $\cos x$ from equation including $\sqrt{\sin x}$ or $\frac{1}{\sqrt{\sin x}}$ using sensible algebra | M1 | e.g. $-3\cos^2 x \sin^2 x + \frac{1}{2}\cos^4 x = 0$ |
| Use correct formula(s) to express their equation/derivative in terms of one trigonometric function | M1 | Can be awarded before the previous M1. May involve more than one trigonometric term |
| Obtain $7\cos^2 x = 6$, $7\sin^2 x = 1$, or $6\tan^2 x = 1$, or equivalent, and obtain answer $x = 0.388$ | A1 | CAO. Question asks for 3 sf. Ignore additional answers outside $\left(0, \frac{\pi}{2}\right)$. $22.2°$ is A0 |
**Total: 6 marks**
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4 The equation of a curve is $y = \cos ^ { 3 } x \sqrt { \sin x }$. It is given that the curve has one stationary point in the interval $0 < x < \frac { 1 } { 2 } \pi$.
Find the $x$-coordinate of this stationary point, giving your answer correct to 3 significant figures.\\
\hfill \mbox{\textit{CAIE P3 2022 Q4 [6]}}