Standard +0.8 This question requires applying the quotient rule to a logarithmic function, setting the derivative equal to zero, and solving the resulting equation ln(x) = 1 to find x = e. While the differentiation is straightforward, recognizing that the stationary point occurs at x = e requires some algebraic manipulation and understanding of logarithms beyond routine application. The 'exact coordinates' requirement adds slight complexity as students must work with e rather than decimal approximations.
Use quotient rule to find first derivative or equivalent
*M1
Obtain \(\frac{dy}{dx} = \frac{3\ln x - 3x \times \frac{1}{x}}{(\ln x)^2}\) or equivalent
A1
Condone lack of brackets in denominator unless specifically simplified to \(2\ln x\)
Equate first derivative to zero and attempt value of \(x\) from \(\ln x = k\)
DM1
Must get as far as \(x =\)
Obtain \(x = e\)
A1
Allow \(e^1\)
Obtain \(y = 3e\)
A1
Allow \(3e^1\); SC1: If \(3\ln x - 3x \times \frac{1}{x} = 0\) seen with no reference to \(\frac{dy}{dx}\), then allow M1 A1 then following marks; SC2: If denominator incorrect and numerator correct/reversed/added then max marks M0A0M1A1A1; SC3: If numerator reversed then max marks M1A0M1A1A1
## Question 3:
| Answer | Mark | Guidance |
|--------|------|----------|
| Use quotient rule to find first derivative or equivalent | *M1 | |
| Obtain $\frac{dy}{dx} = \frac{3\ln x - 3x \times \frac{1}{x}}{(\ln x)^2}$ or equivalent | A1 | Condone lack of brackets in denominator unless specifically simplified to $2\ln x$ |
| Equate first derivative to zero and attempt value of $x$ from $\ln x = k$ | DM1 | Must get as far as $x =$ |
| Obtain $x = e$ | A1 | Allow $e^1$ |
| Obtain $y = 3e$ | A1 | Allow $3e^1$; **SC1**: If $3\ln x - 3x \times \frac{1}{x} = 0$ seen with no reference to $\frac{dy}{dx}$, then allow M1 A1 then following marks; **SC2**: If denominator incorrect and numerator correct/reversed/added then max marks M0A0M1A1A1; **SC3**: If numerator reversed then max marks M1A0M1A1A1 |
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3 Find the exact coordinates of the stationary point of the curve with equation $y = \frac { 3 x } { \ln x }$.\\
\hfill \mbox{\textit{CAIE P2 2019 Q3 [5]}}