11 The function f is defined by \(\mathrm { f } : x \mapsto 6 x - x ^ { 2 } - 5\) for \(x \in \mathbb { R }\).
- Find the set of values of \(x\) for which \(\mathrm { f } ( x ) \leqslant 3\).
- Given that the line \(y = m x + c\) is a tangent to the curve \(y = \mathrm { f } ( x )\), show that \(4 c = m ^ { 2 } - 12 m + 16\).
The function g is defined by \(\mathrm { g } : x \mapsto 6 x - x ^ { 2 } - 5\) for \(x \geqslant k\), where \(k\) is a constant.
- Express \(6 x - x ^ { 2 } - 5\) in the form \(a - ( x - b ) ^ { 2 }\), where \(a\) and \(b\) are constants.
- State the smallest value of \(k\) for which g has an inverse.
- For this value of \(k\), find an expression for \(\mathrm { g } ^ { - 1 } ( x )\).
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