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]