The curve $C$ has equation $y = \frac{ax^2 + x}{x + b},$ where $a$ and $b$ are positive

Question

The curve $C$ has equation $y = \frac{ax^2 + x}{x + b},$ where $a$ and $b$ are positive constants such that $a \neq \frac{1}{b}$. Given that $C$ has no stationary points, use differentiation to find the relationship between $a$ and $b$.[4] It is now given that $a = 2$ and $b = 1$. Sketch $C$, stating the equations of any asymptotes, coordinates of any turning points and of the points where $C$ crosses the axes.[4] Hence find the volume of the solid formed when the finite region bounded by $C$ and the $x$-axis is rotated completely about the $x$-axis.[2]

Accepted Answer

(a) ${0 < ab < 1}$ (b) ${x = -1}$, ${y = 2x - 1}$ ; Turning points ${= (-0.293, -0.172)}$ ; ${(0, 0)}$, ${\left(-\frac{1}{2}, 0\right)}$, ${(-1.71, -5.83)}$ (c) ${0.0245}$ ($3$ s.f.)