Review question

# Can we sketch the graph of this piecewise function? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R7196

## Question

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 $x=0$, $1$, $2$.

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

3. Calculate and factorise $g'(t)$.

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