The function \(f\) is defined by

\[f(x) = \left\{ {\begin{array}{*{20}{l}} {{x^2} - 2,}&{x < 1} \\ {ax + b,}&{x \geqslant 1} \end{array}} \right.\]

where \(a\) and \(b\) are real constants.

Given that both \(f\) and its derivative are continuous at \(x = 1\), find the value of \(a\) and the value of \(b\).

considering continuity \(\mathop {\lim }\limits_{x \to {1^ - }} ({x^2} - 2) =  - 1\)     (M1)

\(a + b =  - 1\)     (A1)

considering differentiability \(2x = a\) when \(x = 1\)     (M1)

\( \Rightarrow a = 2\)     A1

\(b =  - 3\)     A1

[5 marks]