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 \(x=0\), \(1\), \(2\).
Are there any turning points for \(f(x)\)?
- 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?
- Calculate and factorise \(g'(t)\).
What happens if we differentiate \(g(t)\)?
- 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?
