4 Fig. 10 shows a sketch of the graph of \(y = 7 x - x ^ { 2 } - 6\).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{12e190fc-437f-499d-9c27-da49a7546755-2_604_912_1100_638}
\captionsetup{labelformat=empty}
\caption{Fig. 10}
\end{figure}
- Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence find the equation of the tangent to the curve at the point on the curve where \(x = 2\).
Show that this tangent crosses the \(x\)-axis where \(x = \frac { 2 } { 3 }\).
- Show that the curve crosses the \(x\)-axis where \(x = 1\) and find the \(x\)-coordinate of the other point of intersection of the curve with the \(x\)-axis.
- Find \(\int _ { 1 } ^ { 2 } \left( 7 x - x ^ { 2 } - 6 \right) \mathrm { d } x\).
Hence find the area of the region bounded by the curve, the tangent and the \(x\)-axis, shown shaded on Fig. 10.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{12e190fc-437f-499d-9c27-da49a7546755-3_643_1034_267_549}
\captionsetup{labelformat=empty}
\caption{Fig. 11}
\end{figure}
The equation of the curve shown in Fig. 11 is \(y = x ^ { 3 } - 6 x + 2\). - Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
- Find, in exact form, the range of values of \(x\) for which \(x ^ { 3 } - 6 x + 2\) is a decreasing function.
- Find the equation of the tangent to the curve at the point \(( - 1,7 )\).
Find also the coordinates of the point where this tangent crosses the curve again.