7 The function f is defined for all real numbers by
$$f ( x ) = \sin \left( x + \frac { \pi } { 6 } \right)$$
- Find the general solution of the equation \(\mathrm { f } ( x ) = 0\).
- The quadratic function g is defined for all real numbers by
$$\mathrm { g } ( x ) = \frac { 1 } { 2 } + \frac { \sqrt { 3 } } { 2 } x - \frac { 1 } { 4 } x ^ { 2 }$$
It can be shown that \(\mathrm { g } ( x )\) gives a good approximation to \(\mathrm { f } ( x )\) for small values of \(x\).
- Show that \(\mathrm { g } ( 0.05 )\) and \(\mathrm { f } ( 0.05 )\) are identical when rounded to four decimal places.
- A chord joins the points on the curve \(y = \mathrm { g } ( x )\) for which \(x = 0\) and \(x = h\). Find an expression in terms of \(h\) for the gradient of this chord.
- Using your answer to part (b)(ii), find the value of \(\mathrm { g } ^ { \prime } ( 0 )\).