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

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

Are there any turning points for \(f(x)\)?

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

Can we helpfully split \(g(t)\) into two integrals?

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

What happens if we differentiate \(g(t)\)?

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

Where could \(g(t)\) attain its minimum and maximum values?