- In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.
A curve \(C\) has equation
$$x = \sin ^ { 2 } 4 y \quad 0 \leqslant y \leqslant \frac { \pi } { 8 } \quad 0 \leqslant x \leqslant 1$$
The point \(P\) with \(x\) coordinate \(\frac { 1 } { 4 }\) lies on \(C\)
- Find the exact \(y\) coordinate of \(P\)
- Find \(\frac { \mathrm { d } x } { \mathrm {~d} y }\)
- Hence show that \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) can be written in the form
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { q + r ( x + s ) ^ { 2 } } }$$
where \(q , r\) and \(s\) are constants to be found.
Using the answer to part (c),
- state the \(x\) coordinate of the point where the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) is a minimum,
- state the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at this point.