Differentiation

(i) The cost function $C$ , in thousands of pounds, for a specialized manufacturing process is modeled by

\frac{(2x + 5)^2}{3x - 1}, \quad x \neq \frac{1}{3}

where $x\,$ represents the amount of catalyst used in kilograms.

Find $C^{'} (x)$ in the form $P(x)Q(x)\displaystyle \frac{P(x)}{Q(x)}$ where $P (x)$ and $Q (x)$ are fully factorised quadratic expressions.

[5]

Determine the range of values of $x\,$ for which the cost function $C (x)$ is increasing.

[2]

ii.

The vertical displacement $h\,$ of a floating buoy, in metres, is modeled by the function

t\sqrt{\sin 6t}, \quad 0 \le t < \frac{\pi}{6}

where $t\,$ is the time in seconds after a wave passes. The buoy reaches its maximum height at a point $M$ .

Show that the $t$ -coordinate of $M\,$ satisfies the equation $tan⁡6t+kt=0\tan 6t + kt = 0$ , where $k\,$ is a constant to be found.

[5]