Let \[\quad f(x) = \begin{cases}x+1 & \text{for } 0 \leq x \le 1; \\ 2x^2-6x+6 & \text{for } 1 \leq x \le 2. \end{cases}\]

Sketch a graph of \(y=f(x)\) for \(0 \le x \le 2\), labelling any turning points and the values attained at \(x=0\), \(1\), \(2\).

For \(1 \le t \le 2\), define \[g(t) = \int_{t-1}^{t} f(x) \ dx.\] Express \(g(t)\) as a cubic in \(t\).

Calculate and factorise \(g'(t)\).

What are the minimum and maximum values of \(g(t)\) for \(t\) in the range \(1 \le t \le 2\)?