Moderate -0.8 This is a straightforward application of basic differentiation requiring finding dy/dx = 1 - 1/x², solving for stationary points, and using the second derivative test. It's simpler than average A-level questions as it involves only standard techniques with no problem-solving insight needed, though it does require two steps (finding the turning point and classifying it).
4 A curve has equation \(y = x + \frac { 1 } { x }\).
Use calculus to show that the curve has a turning point at \(x = 1\).
Show also that this point is a minimum.
\(1-x^2\) is acceptable Or solving \(1-x^2=0\) to obtain \(x = 1\) or checking \(y'\) before and after \(x = 1\) Valid conclusion First quadrant sketch scores B2
5
$x + x^3$ soi
$y' = 1 - 1/x^2$
subs $x = 1$ to get $y' = 0$
$y''=2x^3$ attempted
Stating $y'' > 0$ so min cao | $1-x^2$ is acceptable Or solving $1-x^2=0$ to obtain $x = 1$ or checking $y'$ before and after $x = 1$ Valid conclusion First quadrant sketch scores B2 | 5
4 A curve has equation $y = x + \frac { 1 } { x }$.\\
Use calculus to show that the curve has a turning point at $x = 1$.\\
Show also that this point is a minimum.
\hfill \mbox{\textit{OCR MEI C2 2005 Q4 [5]}}