The diagram shows the section of the curve $y = \frac{8}{x} + 2x$ for $x > 0$, together with the minimum point $M$.
(a)[5]
Find the expressions for $\frac{dy}{dx}$, $\frac{d^2 y}{dx^2}$ and $\int y^2\,dx$.
(b)[5]
Find the coordinates of $M$, and determine both the coordinates and the nature of the stationary point on the part of the curve where $x < 0$.
(c)[2]
Find the volume produced when the region enclosed by the curve, the $x$-axis and the lines $x = 1$ and $x = 2$ is rotated through $360^\circ$ about the $x$-axis.
Worked solution & mark scheme
This 12-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Correctly obtained derivatives $\frac{dy}{dx}=-8x^{-2}+2$ and $\frac{d^2y}{dx^2}=16x^{-3}$” …