Find stationary points - polynomial/exponential products

11
\includegraphics[max width=\textwidth, alt={}, center]{2d5f452d-f820-40fc-9e22-9d3ac4f0698b-18_661_698_260_717} The diagram shows part of the curve \(y = 3 \sqrt { } ( 4 x + 1 ) - 2 x\). The curve crosses the \(y\)-axis at \(A\) and the stationary point on the curve is \(M\).

1. Obtain expressions for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\int y \mathrm {~d} x\).
2. Find the coordinates of \(M\).
3. Find, showing all necessary working, the area of the shaded region.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

2 Find the exact coordinates of the stationary point on the curve with equation \(y = 5 x \mathrm { e } ^ { \frac { 1 } { 2 } x }\).

6 It is given that the curve \(y = ( x - 2 ) \mathrm { e } ^ { x }\) has one stationary point.

1. Find the exact coordinates of this point.
2. Determine whether this point is a maximum or a minimum point.

3 The curve with equation \(y = 6 \mathrm { e } ^ { x } - \mathrm { e } ^ { 3 x }\) has one stationary point.

1. Find the \(x\)-coordinate of this point.
2. Determine whether this point is a maximum or a minimum point.

3 The equation of a curve is \(y = \frac { 1 } { 2 } \mathrm { e } ^ { 2 x } - 5 \mathrm { e } ^ { x } + 4 x\). Find the exact \(x\)-coordinate of each of the stationary points of the curve and determine the nature of each stationary point.

1. Find, using calculus, the \(x\) coordinate of the stationary point on the curve with equation

$$y = ( 2 x + 5 ) e ^ { 3 x }$$

11. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve \(C\) with equation $$y = \mathrm { e } ^ { a - 3 x } - 3 \mathrm { e } ^ { - x } , \quad x \in \mathbb { R }$$ where \(a\) is a constant and \(a > \ln 4\) The curve \(C\) has a turning point \(P\) and crosses the \(x\)-axis at the point \(Q\) as shown in Figure 2.

1. Find, in terms of \(a\), the coordinates of the point \(P\).
2. Find, in terms of \(a\), the \(x\) coordinate of the point \(Q\).
3. Sketch the curve with equation $$y = \left| \mathrm { e } ^ { a - 3 x } - 3 \mathrm { e } ^ { - x } \right| , \quad x \in \mathbb { R } , \quad a > \ln 4$$ Show on your sketch the exact coordinates, in terms of \(a\), of the points at which the curve meets or cuts the coordinate axes.

1 Fig. 8 shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = ( 1 - x ) \mathrm { e } ^ { 2 x }\), with its turning point P . \begin{figure}[h] \end{figure}

1. Write down the coordinates of the intercepts of \(y = \mathrm { f } ( x )\) with the \(x\) - and \(y\)-axes.
2. Find the exact coordinates of the turning point P .
3. Show that the exact area of the region enclosed by the curve and the \(x\) - and \(y\)-axes is \(\frac { 1 } { 4 } \left( \mathrm { e } ^ { 2 } - 3 \right)\). The function \(\mathrm { g } ( x )\) is defined by \(\mathrm { g } ( x ) = 3 \mathrm { f } \left( \frac { 1 } { 2 } x \right)\).
4. Express \(\mathrm { g } ( x )\) in terms of \(x\). Sketch the curve \(y = \mathrm { g } ( x )\) on the copy of Fig. 8, indicating the coordinates of its intercepts with the \(x\) - and \(y\)-axes and of its turning point.
5. Write down the exact area of the region enclosed by the curve \(y = \mathrm { g } ( x )\) and the \(x\) - and \(y\)-axes.

2 Fig. 9 shows the curve \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = \left( \mathrm { e } ^ { x } - 2 \right) ^ { 2 } - 1 , x \in \mathbb { R } .$$ The curve crosses the \(x\)-axis at O and P , and has a turning point at Q . \begin{figure}[h] \end{figure}

1. Find the exact \(x\)-coordinate of P .
2. Show that the \(x\)-coordinate of Q is \(\ln 2\) and find its \(y\)-coordinate.
3. Find the exact area of the region enclosed by the curve and the \(x\)-axis. The domain of \(\mathrm { f } ( x )\) is now restricted to \(x \geqslant \ln 2\).
4. Find the inverse function \(\mathrm { f } ^ { - 1 } ( x )\). Write down its domain and range, and sketch its graph on the copy of Fig. 9.

5
\includegraphics[max width=\textwidth, alt={}, center]{00a4be37-c095-4d9c-a1cd-d03b8ab1d411-2_455_643_1327_694} The diagram shows the curve \(y = \mathrm { e } ^ { 3 x } - 6 \mathrm { e } ^ { 2 x } + 32\).

1. Find the exact \(x\)-coordinate of the minimum point and verify that the \(y\)-coordinate of the minimum point is 0 .
2. Find the exact area of the region (shaded in the diagram) enclosed by the curve and the axes.

8 Fig. 8 shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = ( 1 - x ) \mathrm { e } ^ { 2 x }\), with its turning point P . \begin{figure}[h] \end{figure}

1. Write down the coordinates of the intercepts of \(y = \mathrm { f } ( x )\) with the \(x\) - and \(y\)-axes.
2. Find the exact coordinates of the turning point P .
3. Show that the exact area of the region enclosed by the curve and the \(x\) - and \(y\)-axes is \(\frac { 1 } { 4 } \left( e ^ { 2 } - 3 \right)\). The function \(\mathrm { g } ( x )\) is defined by \(\mathrm { g } ( x ) = 3 \mathrm { f } \left( \frac { 1 } { 2 } x \right)\).
4. Express \(\mathrm { g } ( x )\) in terms of \(x\). Sketch the curve \(y = \mathrm { g } ( x )\) on the copy of Fig. 8, indicating the coordinates of its intercepts with the \(x\) - and \(y\)-axes and of its turning point.
5. Write down the exact area of the region enclosed by the curve \(y = \mathrm { g } ( x )\) and the \(x\)-and \(y\)-axes.

9 Fig. 9 shows the curve \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = \left( \mathrm { e } ^ { x } - 2 \right) ^ { 2 } - 1 , x \in \mathbb { R } .$$ The curve crosses the \(x\)-axis at O and P , and has a turning point at Q . \begin{figure}[h] \end{figure}

1. Find the exact \(x\)-coordinate of P .
2. Show that the \(x\)-coordinate of Q is \(\ln 2\) and find its \(y\)-coordinate.
3. Find the exact area of the region enclosed by the curve and the \(x\)-axis. The domain of \(\mathrm { f } ( x )\) is now restricted to \(x \geqslant \ln 2\).
4. Find the inverse function \(\mathrm { f } ^ { - 1 } ( x )\). Write down its domain and range, and sketch its graph on the copy of Fig. 9.

9 Fig. 9 shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \mathrm { e } ^ { 2 x } + k \mathrm { e } ^ { - 2 x }\) and \(k\) is a constant greater than 1 . The curve crosses the \(y\)-axis at P and has a turning point Q . \begin{figure}[h] \end{figure}

1. Find the \(y\)-coordinate of P in terms of \(k\).
2. Show that the \(x\)-coordinate of Q is \(\frac { 1 } { 4 } \ln k\), and find the \(y\)-coordinate in its simplest form.
3. Find, in terms of \(k\), the area of the region enclosed by the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = \frac { 1 } { 2 } \ln k\). Give your answer in the form \(a k + b\). The function \(\mathrm { g } ( x )\) is defined by \(\mathrm { g } ( x ) = \mathrm { f } \left( x + \frac { 1 } { 4 } \ln k \right)\).
4. (A) Show that \(\mathrm { g } ( x ) = \sqrt { k } \left( \mathrm { e } ^ { 2 x } + \mathrm { e } ^ { - 2 x } \right)\).
   (B) Hence show that \(\mathrm { g } ( x )\) is an even function.
   (C) Deduce, with reasons, a geometrical property of the curve \(y = \mathrm { f } ( x )\). \section*{END OF QUESTION PAPER}