Problem

The equation of a curve is $y = kx^{\frac{1}{2}} - 4x^2 + 2$, where $k$ is a constant.

(a) Find $\frac{dy}{dx}$ and $\frac{d^2y}{dx^2}$ in terms of $k$.

(b) It is given that $k = 2$.

Find the coordinates of the stationary point and determine its nature.

(c) Points $A$ and $B$ on the curve have $x$-coordinates 0.25 and 1 respectively. For a different value of $k$, the tangents to the curve at the points $A$ and $B$ meet at a point with $x$-coordinate 0.6.

Find this value of $k$.