The curve is given by $y = \frac{1}{k}x^{\frac{1}{2}} + x^{-\frac{1}{2}} + \frac{1}{k^2}$ for $x > 0$, with $k$ a positive constant.
(a)[4]
At $x = \frac{1}{4}$, the gradient of the curve is $3$. Find the value of $k$.
(b)[5]
Instead, it is given that $\int_{\frac{1}{4k^2}}^{k^2} \left( \frac{1}{k}x^{\frac{1}{2}} + x^{-\frac{1}{2}} + \frac{1}{k^2} \right) \, dx = \frac{13}{12}$. Find the value of $k$.
Worked solution & mark scheme
This 9-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Obtains the correct derivative” …