Standard +0.8 This is a Further Maths question requiring implicit differentiation of a rational function using the quotient rule, solving an inequality involving a discriminant (no stationary points means dy/dx ≠ 0), then analyzing asymptotes (vertical, oblique via polynomial division) and sketching. The multi-step nature, discriminant analysis, and oblique asymptote identification elevate this above standard C3/C4 calculus questions, though it follows established techniques without requiring novel insight.
The curve \(C\) has equation \(y = \frac{2x^2 + kx}{x + 1}\), where \(k\) is a constant. Find the set of values of \(k\) for which \(C\) has no stationary points. [5]
For the case \(k = 4\), find the equations of the asymptotes of \(C\) and sketch \(C\), indicating the coordinates of the points where \(C\) intersects the coordinate axes. [6]
The curve $C$ has equation $y = \frac{2x^2 + kx}{x + 1}$, where $k$ is a constant. Find the set of values of $k$ for which $C$ has no stationary points. [5]
For the case $k = 4$, find the equations of the asymptotes of $C$ and sketch $C$, indicating the coordinates of the points where $C$ intersects the coordinate axes. [6]
\hfill \mbox{\textit{CAIE FP1 2015 Q8 [11]}}