3. Let

$$f ( x ) = \left\{ \begin{array} { c c } 
x + 1 & \text { for } 0 \leq x \leq 1 \\
2 x ^ { 2 } - 6 x + 6 & \text { for } 1 \leq x \leq 2
\end{array} \right.$$

(a) On the axes provided below, sketch a graph of $y = f ( x )$ for $0 \leq x \leq 2$, labelling any turning points and the values attained at $x = 0,1,2$.\\
(b) For $1 \leq t \leq 2$, define

$$g ( t ) = \int _ { t - 1 } ^ { t } f ( x ) \mathrm { d } x$$

Express $g ( t )$ as a cubic in $t$.\\
(c) Calculate and factorize $g ^ { \prime } ( t )$.\\
(d) What are the minimum and maximum values of $g ( t )$ for $t$ in the range $1 \leq t \leq 2$ ?\\
\includegraphics[max width=\textwidth, alt={}, center]{1f226339-1d44-461f-a995-fafb503da240-10_1262_780_1283_550}\\
4.\\
\includegraphics[max width=\textwidth, alt={}, center]{1f226339-1d44-461f-a995-fafb503da240-12_504_595_397_296}\\
\includegraphics[max width=\textwidth, alt={}, center]{1f226339-1d44-461f-a995-fafb503da240-12_501_595_397_1016}

The triangle $A B C$, drawn above, has sides $B C , C A$ and $A B$ of length $a , b$ and $c$ respectively, and the angles at $A , B$ and $C$ are $\alpha , \beta$ and $\gamma$.\\
(a) Show that the area of $A B C$ equals $\frac { 1 } { 2 } b c \sin \alpha$.

Deduce the sine rule

$$\frac { a } { \sin \alpha } = \frac { b } { \sin \beta } = \frac { c } { \sin \gamma } .$$

(b) In the triangle above, let $P , Q$ and $R$ respectively be the feet of the perpendiculars from $A$ to $B C , B$ to $C A$, and $C$ to $A B$, as shown.

Prove that

$$\text { Area of } P Q R = \left( 1 - \cos ^ { 2 } \alpha - \cos ^ { 2 } \beta - \cos ^ { 2 } \gamma \right) \times ( \text { Area of } A B C ) .$$

For what triangles $A B C$, with angles $\alpha , \beta , \gamma$, does the equation

$$\cos ^ { 2 } \alpha + \cos ^ { 2 } \beta + \cos ^ { 2 } \gamma = 1$$

hold?\\