Moderate -0.3 This is a straightforward application of the quotient/chain rule to find dy/dx, substitute x=2 to verify dy/dx=0, then use the second derivative test. The algebra is simple with small numbers, and 'verify' means students know the answer in advance. Slightly easier than average due to the verification format removing problem-solving uncertainty.
5 A curve has equation \(y = 2 x + \frac { 1 } { ( x - 1 ) ^ { 2 } }\). Verify that the curve has a stationary point at \(x = 2\) and determine its nature.
Correct \(\frac{d^2y}{dx^2}\) and 'minimum' is required. Or other valid method for last 2 marks
$\frac{dy}{dx} = 2 - 2(x-1)^{-3}$ | B2,1,0 | $-1$ each error in 2, $-2$, $(x-1)^{-3}$
Sub $x = 2 \to \frac{dy}{dx} = 2 - 2 = 0 \Rightarrow$ stat value at $x = 2$ | B1 | AG
$\frac{d^2y}{dx^2} = 6(x-1)^{-4}$ (and sub $x = 2$) | M1 | Reasonable attempt to diff form $(x-1)^{-n}$
(At $x = 2, \frac{d^2y}{dx^2} = 6$) $> 0 \Rightarrow$ Minimum | A1 | Correct $\frac{d^2y}{dx^2}$ and 'minimum' is required. Or other valid method for last 2 marks | [5]
5 A curve has equation $y = 2 x + \frac { 1 } { ( x - 1 ) ^ { 2 } }$. Verify that the curve has a stationary point at $x = 2$ and determine its nature.
\hfill \mbox{\textit{CAIE P1 2012 Q5 [5]}}