The function \(f\) is defined by

\[f\left( x \right) = \left\{ {\begin{array}{*{20}{c}}
{\left| {x - 2} \right| + 1}&{x < 2} \\
{a{x^2} + bx}&{x \geqslant 2}
\end{array}} \right.\]

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

Given that both \(f\) and its derivative are continuous at \(x = 2\), find the value of \(a\) and the value of \(b\).

considering continuity at \(x = 2\)

\(\mathop {{\text{lim}}}\limits_{x \to {2^ - }} f\left( x \right) = 1\) and \(\mathop {{\text{lim}}}\limits_{x \to {2^ + }} f\left( x \right) = 4a + 2b\)    (M1)

\(4a + 2b = 1\)     A1

considering differentiability at \(x = 2\)

\(f'\left( x \right) = \left\{ {\begin{array}{*{20}{c}}
{ - 1}&{x < 2} \\
{2ax + b}&{x \geqslant 2}
\end{array}} \right.\)    (M1)

\(\mathop {{\text{lim}}}\limits_{x \to {2^ - }} f'\left( x \right) =  - 1\) and \(\mathop {{\text{lim}}}\limits_{x \to {2^ + }} f'\left( x \right) = 4a + b\)     (M1)

Note: The above M1 is for attempting to find the left and right limit of their derived piecewise function at \(x = 2\).

\(4a + b =  - 1\)     A1

\(a =  - \frac{3}{4}\) and \(b = 2\)     A1

[6 marks]