Find stationary points and nature

5 The equation of a curve is \(y = 2 x ^ { 2 } - \frac { 1 } { 2 x } + 3\).

1. Find the coordinates of the stationary point.
2. Determine the nature of the stationary point.
3. For positive values of \(x\), determine whether the curve shows a function that is increasing, decreasing or neither. Give a reason for your answer.

8 The equation of a curve is \(y = 2 x + 1 + \frac { 1 } { 2 x + 1 }\) for \(x > - \frac { 1 } { 2 }\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
2. Find the coordinates of the stationary point and determine the nature of the stationary point.

6 The non-zero variables \(x , y\) and \(u\) are such that \(u = x ^ { 2 } y\). Given that \(y + 3 x = 9\), find the stationary value of \(u\) and determine whether this is a maximum or a minimum value.

5 A curve has equation \(y = \frac { 1 } { x - 3 } + x\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
2. Find the coordinates of the maximum point \(A\) and the minimum point \(B\) on the curve.

9 A curve has equation \(y = \frac { k ^ { 2 } } { x + 2 } + x\), where \(k\) is a positive constant. Find, in terms of \(k\), the values of \(x\) for which the curve has stationary points and determine the nature of each stationary point.

5 A curve has equation \(y = \frac { 8 } { x } + 2 x\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
2. Find the coordinates of the stationary points and state, with a reason, the nature of each stationary point.

5 A curve has equation \(y = \frac { 8 } { x } + 2 x\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
2. Find the coordinates of the stationary points and state, with a reason, the nature of each stationary point.

4 The curve with equation \(y = \frac { \mathrm { e } ^ { 2 x } } { x ^ { 3 } }\) has one stationary point.

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

4 The curve with equation \(y = \frac { 2 - \sin x } { \cos x }\) has one stationary point in the interval \(- \frac { 1 } { 2 } \pi < x < \frac { 1 } { 2 } \pi\).

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 \(y = x \mathrm { e } ^ { 1 - 2 x }\) has one stationary point.

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

3. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = ( x - 2 ) ^ { 2 } \mathrm { e } ^ { 3 x } \quad x \in \mathbb { R }$$ The curve has a maximum turning point at \(A\) and a minimum turning point at \(( 2,0 )\)

1. Use calculus to find the exact coordinates of \(A\). Given that the equation \(\mathrm { f } ( x ) = k\), where \(k\) is a constant, has at least two distinct roots,
2. state the range of possible values for \(k\).

3. A curve \(C\) has equation $$y = x ^ { 2 } \mathrm { e } ^ { x }$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), using the product rule for differentiation.
2. Hence find the coordinates of the turning points of \(C\).
3. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
4. Determine the nature of each turning point of the curve \(C\).

4. $$\mathrm { f } ( x ) = 25 x ^ { 2 } \mathrm { e } ^ { 2 x } - 16 , \quad x \in \mathbb { R }$$

1. Using calculus, find the exact coordinates of the turning points on the curve with equation \(y = \mathrm { f } ( x )\).
2. Show that the equation \(\mathrm { f } ( x ) = 0\) can be written as \(x = \pm \frac { 4 } { 5 } \mathrm { e } ^ { - x }\) The equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\), where \(\alpha = 0.5\) to 1 decimal place.
3. Starting with \(x _ { 0 } = 0.5\), use the iteration formula $$x _ { n + 1 } = \frac { 4 } { 5 } \mathrm { e } ^ { - x _ { n } }$$ to calculate the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\), giving your answers to 3 decimal places.
4. Give an accurate estimate for \(\alpha\) to 2 decimal places, and justify your answer.

8 A curve has equation \(y = ( x + 2 ) \mathrm { e } ^ { - x }\).

1. Find the coordinates of the points where the curve cuts the axes.
2. Find the coordinates of the stationary point, S , on the curve.
3. By evaluating \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at S , determine whether the stationary point is a maximum or a minimum.
4. Sketch the curve in the domain \(- 3 < x < 3\).
5. Find where the normal to the curve at the point \(( 0,2 )\) cuts the curve again.
6. Find the area of the region bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 3\).

8 Fig. 8 shows part of the graph of the function \(y = 5 x ( 2 x - 1 ) ^ { 3 }\). \begin{figure}[h] \end{figure}

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence find the \(x\)-coordinate of S , the turning point of the curve.
2. Find the area of the shaded region enclosed between the curve and the \(x\)-axis.
3. Given that \(\mathrm { f } ( x ) = 5 x ( 2 x - 1 ) ^ { 3 }\), show that \(\mathrm { f } ( x + 0.5 ) = 40 x ^ { 3 } ( x + 0.5 )\).
4. Find \(\int _ { - \frac { 1 } { 2 } } ^ { 0 } 40 x ^ { 3 } ( x + 0.5 ) \mathrm { d } x\).
5. Explain, with the aid of a sketch, the connection between your answer to parts (ii) and (iv).

1. (a) The function \(\mathrm { f } ( x )\) has \(\mathrm { f } ^ { \prime } ( x ) = \frac { \mathrm { u } ( x ) } { \mathrm { v } ( x ) }\). Given that \(\mathrm { f } ^ { \prime } ( k ) = 0\), show that \(\mathrm { f } ^ { \prime \prime } ( k ) = \frac { \mathrm { u } ^ { \prime } ( k ) } { \mathrm { v } ( k ) }\).

\begin{figure}[h] \end{figure} (b) The curve \(C\) with equation $$y = \frac { 2 x ^ { 2 } + 3 } { x ^ { 2 } - 1 }$$ crosses the \(y\)-axis at the point \(A\). Figure 1 shows a sketch of \(C\) together with its 3 asymptotes.

1. Find the coordinates of the point \(A\).
2. Find the equations of the asymptotes of \(C\). The point \(P ( a , b ) , a > 0\) and \(b > 0\), lies on \(C\). The point \(Q\) also lies on \(C\) with \(P Q\) parallel to the \(x\)-axis and \(A P = A Q\).
3. Show that the area of triangle \(P A Q\) is given by \(\frac { 5 a ^ { 3 } } { a ^ { 2 } - 1 }\).
4. Find, as \(a\) varies, the minimum area of triangle \(P A Q\), giving your answer in its simplest form.

9. \section*{Figure 2} Figure 2 shows a sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = 4 \left( x ^ { 2 } - 2 \right) \mathrm { e } ^ { - 2 x } \quad x \in \mathbb { R }$$

1. Show that \(\mathrm { f } ^ { \prime } ( x ) = 8 \left( 2 + x - x ^ { 2 } \right) \mathrm { e } ^ { - 2 x }\)
2. Hence find, in simplest form, the exact coordinates of the stationary points of \(C\). The function g and the function h are defined by $$\begin{array} { l l } \mathrm { g } ( x ) = 2 \mathrm { f } ( x ) & x \in \mathbb { R } \\ \mathrm {~h} ( x ) = 2 \mathrm { f } ( x ) - 3 & x \geqslant 0 \end{array}$$
3. Find (i) the range of \(g\) (ii) the range of h

14 The equation of a curve is \(y = x ^ { 2 } ( x - 2 ) ^ { 3 }\).

1. Find \(\frac { \mathrm { dy } } { \mathrm { dx } }\), giving your answer in factorised form.
2. Determine the coordinates of the stationary points on the curve. In part (c) you may use the result \(\frac { d ^ { 2 } y } { d x ^ { 2 } } = 4 ( x - 2 ) \left( 5 x ^ { 2 } - 8 x + 2 \right)\).
3. Determine the nature of the stationary points on the curve.
4. Sketch the curve.

12

1. Show that the only stationary point on the curve \(\mathrm { y } = \frac { \ln \mathrm { x } } { \mathrm { x } }\) occurs where \(x = \mathrm { e }\), as given in line 45.
2. Show that the stationary point is a maximum.
3. It follows from part (b) that, for any positive number \(a\) with \(a \neq \mathrm { e }\), \(\frac { \ln \mathrm { e } } { \mathrm { e } } > \frac { \ln a } { a }\).
   Use this fact to show that \(\mathrm { e } ^ { a } > a ^ { \mathrm { e } }\).

6 The diagram shows the curve \(y = \frac { \ln x } { x }\). \includegraphics[max width=\textwidth, alt={}, center]{33ca7e6d-b9eb-46be-b5b0-c5685212d7ff-4_586_1034_1612_513} The curve crosses the \(x\)-axis at \(A\) and has a stationary point at \(B\).

1. State the coordinates of \(A\).
2. Find the coordinates of the stationary point, \(B\), of the curve, giving your answer in an exact form.
3. Find the exact value of the gradient of the normal to the curve at the point where \(x = \mathrm { e } ^ { 3 }\).

4. The curve with equation \(y = x ^ { \frac { 5 } { 2 } } \ln \frac { x } { 4 } , x > 0\) crosses the \(x\)-axis at the point \(P\).

1. Write down the coordinates of \(P\). The normal to the curve at \(P\) crosses the \(y\)-axis at the point \(Q\).
2. Find the area of triangle \(O P Q\) where \(O\) is the origin. The curve has a stationary point at \(R\).
3. Find the \(x\)-coordinate of \(R\) in exact form.

6. A curve has the equation \(y = \mathrm { e } ^ { 3 x } \cos 2 x\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \mathrm { e } ^ { 3 x } ( 5 \cos 2 x - 12 \sin 2 x )\). The curve has a stationary point in the interval \([ 0,1 ]\).
3. Find the \(x\)-coordinate of the stationary point to 3 significant figures.
4. Determine whether the stationary point is a maximum or minimum point and justify your answer.

7. The curve \(C\) has equation \(y = \frac { x } { 4 + x ^ { 2 } }\).

1. Use calculus to find the coordinates of the turning points of \(C\). Using the result \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { 2 x \left( x ^ { 2 } - 12 \right) } { \left( 4 + x ^ { 2 } \right) ^ { 3 } }\), or otherwise,
2. determine the nature of each of the turning points.
3. Sketch the curve \(C\).

6 A curve has equation \(y = \mathrm { e } ^ { 2 x } \left( x ^ { 2 } - 4 x - 2 \right)\).

1. Find the value of the \(x\)-coordinate of each of the stationary points of the curve.
2. 1. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
   2. Determine the nature of each of the stationary points of the curve.

1 A curve has equation \(y = \mathrm { e } ^ { - 4 x } \left( x ^ { 2 } + 2 x - 2 \right)\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 \mathrm { e } ^ { - 4 x } \left( 5 - 3 x - 2 x ^ { 2 } \right)\).
2. Find the exact values of the coordinates of the stationary points of the curve.