1.07n Stationary points: find maxima, minima using derivatives

9 The equation of a curve is \(y = 12 x - 4 x ^ { \frac { 3 } { 2 } }\).

1. State the coordinates of the intersection of the curve with the \(y\)-axis.
2. Find the value of \(y\) when \(x = 9\).
3. Determine the coordinates of the stationary point.
4. Sketch the curve, giving the coordinates of the stationary point and of any intercepts with the axes.

11 In this question you must show detailed reasoning.
The equation of a curve is \(y = 2 x ^ { 3 } + 9 x ^ { 2 } + 24 x - 8\).
Show that there are no stationary points on this curve.

9 The equation of a curve is \(y = 24 \sqrt { x } - 8 x ^ { \frac { 3 } { 2 } } + 16\).

1. Find \(\frac { \mathrm { dy } } { \mathrm { dx } }\).
2. Find the coordinates of the turning point.
3. Determine the nature of the turning point.

13 The curve with equation \(\mathrm { y } = \mathrm { px } + \frac { 8 } { \mathrm { x } ^ { 2 } } + \mathrm { q }\), where \(p\) and \(q\) are constants, has a stationary point at \(( 2,7 )\).

1. Determine the values of \(p\) and \(q\).
2. Find \(\frac { d ^ { 2 } y } { d x ^ { 2 } }\).
3. Hence determine the nature of the stationary point at (2, 7).

16 A block of mass \(m\) kg rests on rough horizontal ground. The coefficient of friction between the block and the ground is \(\mu\). A force of magnitude \(T \mathrm {~N}\) is applied at an angle \(\theta\) radians above the horizontal as shown in the diagram and the block slides without tilting or lifting. \includegraphics[max width=\textwidth, alt={}, center]{1d0ca3d5-6529-435f-a0b8-50ea4859adde-10_291_707_388_239}

1. Show that the acceleration of the block is given by \(\frac { T } { m } \cos \theta - \mu g + \frac { T } { m } \mu \sin \theta\). For a fixed value of \(T\), the acceleration of the block depends on the value of \(\theta\). The acceleration has its greatest value when \(\theta = \alpha\).
2. Find an expression for \(\alpha\) in terms of \(\mu\).

12 A function is defined by \(\mathrm { f } ( x ) = x ^ { 3 } - x\).

1. By considering \(\frac { f ( x + h ) - f ( x ) } { h }\), show from first principles that \(f ^ { \prime } ( x ) = 3 x ^ { 2 } - 1\).
2. Sketch the gradient function \(\mathrm { f } ^ { \prime } ( x )\).
3. Show that the curve \(y = f ( x )\) has a single point of inflection which is not a stationary point.

8 The equation of a curve is \(y = 2 x ^ { 3 } + 3 m x ^ { 2 } - 9 m x + 4\). Determine the range of values of \(m\) for which the curve has no stationary values.

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.

4 In this question you must show detailed reasoning.
A curve has equation \(y = x - 5 + \frac { 1 } { x - 2 }\). The curve is shown in Fig. 4. \begin{figure}[h] \end{figure}

1. Determine the coordinates of the stationary points on the curve.
2. Determine the nature of each stationary point.
3. Write down the equation of the vertical asymptote.
4. Deduce the set of values of \(x\) for which the curve is concave upwards.

5 A curve is defined implicitly by the equation \(2 x ^ { 2 } + 3 x y + y ^ { 2 } + 2 = 0\).

1. Show that \(\frac { d y } { d x } = - \frac { 4 x + 3 y } { 3 x + 2 y }\).
2. In this question you must show detailed reasoning. Find the coordinates of the stationary points of the curve.

7 A wire, 10 cm long, is bent to form the perimeter of a sector of a circle, as shown in the diagram. The radius is \(r \mathrm {~cm}\) and the angle at the centre is \(\theta\) radians. \includegraphics[max width=\textwidth, alt={}, center]{20639e13-01cc-4d96-b694-fb3cf1828f4d-07_323_204_342_242} Determine the maximum possible area of the sector, showing that it is a maximum.

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 \(\mathrm { e } ^ { a } > a ^ { \mathrm { e } }\).

7 The volume, \(V \mathrm {~m} ^ { 3 }\), of water in a tank at time \(t\) seconds is given by $$V = \frac { 1 } { 3 } t ^ { 6 } - 2 t ^ { 4 } + 3 t ^ { 2 } , \quad \text { for } t \geqslant 0$$

1. Find:
   1. \(\frac { \mathrm { d } V } { \mathrm {~d} t }\);
      (3 marks)
   2. \(\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} t ^ { 2 } }\).
      (2 marks)
2. Find the rate of change of the volume of water in the tank, in \(\mathrm { m } ^ { 3 } \mathrm {~s} ^ { - 1 }\), when \(t = 2\).
3. 1. Verify that \(V\) has a stationary value when \(t = 1\).
   2. Determine whether this is a maximum or minimum value.

6 The diagram below shows a rectangular sheet of metal 24 cm by 9 cm . \includegraphics[max width=\textwidth, alt={}, center]{10bca9b4-5327-4b35-8b75-612b396e8a76-4_512_897_386_561} A square of side \(x \mathrm {~cm}\) is cut from each corner and the metal is then folded along the broken lines to make an open box with a rectangular base and height \(x \mathrm {~cm}\).

1. Show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of liquid the box can hold is given by $$V = 4 x ^ { 3 } - 66 x ^ { 2 } + 216 x$$
2. 1. Find \(\frac { \mathrm { d } V } { \mathrm {~d} x }\).
   2. Show that any stationary values of \(V\) must occur when \(x ^ { 2 } - 11 x + 18 = 0\).
   3. Solve the equation \(x ^ { 2 } - 11 x + 18 = 0\).
   4. Explain why there is only one value of \(x\) for which \(V\) is stationary.
3. 1. Find \(\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} x ^ { 2 } }\).
   2. Hence determine whether the stationary value is a maximum or minimum.

7 The volume, \(V \mathrm {~m} ^ { 3 }\), of water in a tank at time \(t\) seconds is given by $$V = \frac { 1 } { 3 } t ^ { 6 } - 2 t ^ { 4 } + 3 t ^ { 2 } , \quad \text { for } t \geqslant 0$$

1. Find:
   1. \(\frac { \mathrm { d } V } { \mathrm {~d} t }\);
      (3 marks)
   2. \(\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} t ^ { 2 } }\).
      (2 marks)
2. Find the rate of change of the volume of water in the tank, in \(\mathrm { m } ^ { 3 } \mathrm {~s} ^ { - 1 }\), when \(t = 2\).
   (2 marks)
3. 1. Verify that \(V\) has a stationary value when \(t = 1\).
      (2 marks)
   2. Determine whether this is a maximum or minimum value.
      (2 marks)

5 A model car moves so that its distance, \(x\) centimetres, from a fixed point \(O\) after time \(t\) seconds is given by $$x = \frac { 1 } { 2 } t ^ { 4 } - 20 t ^ { 2 } + 66 t , \quad 0 \leqslant t \leqslant 4$$

1. Find:
   1. \(\frac { \mathrm { d } x } { \mathrm {~d} t }\);
   2. \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } }\).
2. Verify that \(x\) has a stationary value when \(t = 3\), and determine whether this stationary value is a maximum value or a minimum value.
3. Find the rate of change of \(x\) with respect to \(t\) when \(t = 1\).
4. Determine whether the distance of the car from \(O\) is increasing or decreasing at the instant when \(t = 2\).

3 The depth of water, \(y\) metres, in a tank after time \(t\) hours is given by $$y = \frac { 1 } { 8 } t ^ { 4 } - 2 t ^ { 2 } + 4 t , \quad 0 \leqslant t \leqslant 4$$

1. Find:
   1. \(\frac { \mathrm { d } y } { \mathrm {~d} t }\);
   2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } }\).
2. Verify that \(y\) has a stationary value when \(t = 2\) and determine whether it is a maximum value or a minimum value.
3. 1. Find the rate of change of the depth of water, in metres per hour, when \(t = 1\).
   2. Hence determine, with a reason, whether the depth of water is increasing or decreasing when \(t = 1\).

1 The curve with equation \(y = 13 + 18 x + 3 x ^ { 2 } - 4 x ^ { 3 }\) passes through the point \(P\) where \(x = - 1\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Show that the point \(P\) is a stationary point of the curve and find the other value of \(x\) where the curve has a stationary point.
3. 1. Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at the point \(P\).
   2. Hence, or otherwise, determine whether \(P\) is a maximum point or a minimum point.
      (l mark)

4 The curve with equation \(y = x ^ { 5 } - 3 x ^ { 2 } + x + 5\) is sketched below. The point \(O\) is at the origin and the curve passes through the points \(A ( - 1,0 )\) and \(B ( 1,4 )\). \includegraphics[max width=\textwidth, alt={}, center]{91170a77-e266-4c81-89ee-1fc29a538485-3_447_752_438_653}

1. Given that \(y = x ^ { 5 } - 3 x ^ { 2 } + x + 5\), find:
   1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\);
   2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
2. Find an equation of the tangent to the curve at the point \(A ( - 1,0 )\).
3. Verify that the point \(B\), where \(x = 1\), is a minimum point of the curve.
4. The curve with equation \(y = x ^ { 5 } - 3 x ^ { 2 } + x + 5\) is sketched below. The point \(O\) is at the origin and the curve passes through the points \(A ( - 1,0 )\) and \(B ( 1,4 )\). \includegraphics[max width=\textwidth, alt={}, center]{91170a77-e266-4c81-89ee-1fc29a538485-3_451_757_1736_648}
   1. Find \(\int _ { - 1 } ^ { 1 } \left( x ^ { 5 } - 3 x ^ { 2 } + x + 5 \right) \mathrm { d } x\).
   2. Hence find the area of the shaded region bounded by the curve between \(A\) and \(B\) and the line segments \(A O\) and \(O B\).

4 The curve with equation \(y = x ^ { 3 } - 5 x ^ { 2 } + 7 x - 3\) is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{3729de55-7139-4f41-8584-640f173c0e09-3_444_588_411_717} The curve touches the \(x\)-axis at the point \(A ( 1,0 )\) and cuts the \(x\)-axis at the point \(B\).

1. 1. Use the factor theorem to show that \(x - 3\) is a factor of $$\mathrm { p } ( x ) = x ^ { 3 } - 5 x ^ { 2 } + 7 x - 3$$
   2. Hence find the coordinates of \(B\).
2. The point \(M\), shown on the diagram, is a minimum point of the curve with equation \(y = x ^ { 3 } - 5 x ^ { 2 } + 7 x - 3\).
   1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   2. Hence determine the \(x\)-coordinate of \(M\).
3. Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) when \(x = 1\).
4. 1. Find \(\int \left( x ^ { 3 } - 5 x ^ { 2 } + 7 x - 3 \right) \mathrm { d } x\).
   2. Hence determine the area of the shaded region bounded by the curve and the coordinate axes.

5 The curve with equation \(y = x ^ { 3 } - 10 x ^ { 2 } + 28 x\) is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{f2c95d73-d3fe-48f7-af07-84f12bb06727-3_483_899_402_568} The curve crosses the \(x\)-axis at the origin \(O\) and the point \(A ( 3,21 )\) lies on the curve.

1. 1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   2. Hence show that the curve has a stationary point when \(x = 2\) and find the \(x\)-coordinate of the other stationary point.
2. 1. Find \(\int \left( x ^ { 3 } - 10 x ^ { 2 } + 28 x \right) \mathrm { d } x\).
   2. Hence show that \(\int _ { 0 } ^ { 3 } \left( x ^ { 3 } - 10 x ^ { 2 } + 28 x \right) \mathrm { d } x = 56 \frac { 1 } { 4 }\).
   3. Hence determine the area of the shaded region bounded by the curve and the line \(O A\).

6 The diagram shows a block of wood in the shape of a prism with triangular cross-section. The end faces are right-angled triangles with sides of lengths \(3 x \mathrm {~cm}\), \(4 x \mathrm {~cm}\) and \(5 x \mathrm {~cm}\), and the length of the prism is \(y \mathrm {~cm}\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{66813123-3876-4484-aad1-4bfc09bb1508-7_394_825_459_548} The total surface area of the five faces is \(144 \mathrm {~cm} ^ { 2 }\).

1. 1. Show that \(x y + x ^ { 2 } = 12\).
   2. Hence show that the volume of the block, \(V \mathrm {~cm} ^ { 3 }\), is given by $$V = 72 x - 6 x ^ { 3 }$$
2. 1. Find \(\frac { \mathrm { d } V } { \mathrm {~d} x }\).
   2. Show that \(V\) has a stationary value when \(x = 2\).
3. Find \(\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} x ^ { 2 } }\) and hence determine whether \(V\) has a maximum value or a minimum value when \(x = 2\).
   (2 marks)

3 The volume, \(V \mathrm {~m} ^ { 3 }\), of water in a tank after time \(t\) seconds is given by $$V = \frac { t ^ { 3 } } { 4 } - 3 t + 5$$

1. Find \(\frac { \mathrm { d } V } { \mathrm {~d} t }\).
2. 1. Find the rate of change of volume, in \(\mathrm { m } ^ { 3 } \mathrm {~s} ^ { - 1 }\), when \(t = 1\).
   2. Hence determine, with a reason, whether the volume is increasing or decreasing when \(t = 1\).
3. 1. Find the positive value of \(t\) for which \(V\) has a stationary value.
   2. Find \(\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} t ^ { 2 } }\), and hence determine whether this stationary value is a maximum value or a minimum value.
      (3 marks)

4 The diagram shows a solid cuboid with sides of lengths \(x \mathrm {~cm} , 3 x \mathrm {~cm}\) and \(y \mathrm {~cm}\). \includegraphics[max width=\textwidth, alt={}, center]{dbc25177-4a28-480f-93d5-41acb2a2d28c-3_349_472_376_769} The total surface area of the cuboid is \(32 \mathrm {~cm} ^ { 2 }\).

1. 1. Show that \(3 x ^ { 2 } + 4 x y = 16\).
   2. Hence show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the cuboid is given by $$V = 12 x - \frac { 9 x ^ { 3 } } { 4 }$$
2. Find \(\frac { \mathrm { d } V } { \mathrm {~d} x }\).
3. 1. Verify that a stationary value of \(V\) occurs when \(x = \frac { 4 } { 3 }\).
   2. Find \(\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} x ^ { 2 } }\) and hence determine whether \(V\) has a maximum value or a minimum value when \(x = \frac { 4 } { 3 }\).

4

1. The polynomial \(\mathrm { f } ( x )\) is given by \(\mathrm { f } ( x ) = x ^ { 3 } - 4 x + 15\).
   1. Use the Factor Theorem to show that \(x + 3\) is a factor of \(\mathrm { f } ( x )\).
   2. Express \(\mathrm { f } ( x )\) in the form \(( x + 3 ) \left( x ^ { 2 } + p x + q \right)\), where \(p\) and \(q\) are integers.
2. A curve has equation \(y = x ^ { 4 } - 8 x ^ { 2 } + 60 x + 7\).
   1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   2. Show that the \(x\)-coordinates of any stationary points of the curve satisfy the equation $$x ^ { 3 } - 4 x + 15 = 0$$
   3. Use the results above to show that the only stationary point of the curve occurs when \(x = - 3\).
   4. Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) when \(x = - 3\).
   5. Hence determine, with a reason, whether the curve has a maximum point or a minimum point when \(x = - 3\).