The curve is defined by $y = f(x)$, and it is stated that $f'(x) = \left(\frac{1}{2}x + k\right)^{-2} - (1 + k)^{-2}$, where $k$ is a constant. The curve has a minimum point when $x = 2$.
(a)[3]
Determine $f''(x)$ in terms of $k$ and $x$, and hence determine the possible values of $k$.
(b)[4]
It is now stated that $k = -3$ and the minimum point is at $(2, 3\frac{1}{2})$. Find $f(x)$.
(c)[4]
Find the coordinates of the other stationary point and state its nature.
Worked solution & mark scheme
This 11-mark question has a full step-by-step worked solution and mark scheme. One marking point: “The correct second derivative is $f''(x)=-\left(\frac{1}{2}x+k\right)^{-3}$” …