Standard +0.3 This is a straightforward stationary points question requiring quotient rule differentiation of a logarithmic function, setting the derivative to zero, and solving a simple quadratic in ln(x). While it involves transcendental functions, the technique is standard and the algebra is clean, making it slightly easier than average.
4 The curve with equation \(y = \frac { ( \ln x ) ^ { 2 } } { x }\) has two stationary points. Find the exact values of the coordinates of these points.
State or imply derivative of \((\ln x)^2\) is \(\frac{2\ln x}{x}\)
B1
Use correct quotient or product rule
M1
Obtain correct derivative in any form, e.g. \(\frac{2\ln x}{x^2} - \frac{(\ln x)^2}{x^2}\)
A1
Equate derivative (or its numerator) to zero and solve for \(\ln x\)
M1
Obtain the point \((1, 0)\) with no errors seen
A1
Obtain the point \((e^2, 4e^{-2})\)
A1
[6]
State or imply derivative of $(\ln x)^2$ is $\frac{2\ln x}{x}$ | B1 |
Use correct quotient or product rule | M1 |
Obtain correct derivative in any form, e.g. $\frac{2\ln x}{x^2} - \frac{(\ln x)^2}{x^2}$ | A1 |
Equate derivative (or its numerator) to zero and solve for $\ln x$ | M1 |
Obtain the point $(1, 0)$ with no errors seen | A1 |
Obtain the point $(e^2, 4e^{-2})$ | A1 | [6]
4 The curve with equation $y = \frac { ( \ln x ) ^ { 2 } } { x }$ has two stationary points. Find the exact values of the coordinates of these points.
\hfill \mbox{\textit{CAIE P3 2016 Q4 [6]}}