1.07p Points of inflection: using second derivative

8

1. Find the coordinates of the stationary point on the curve \(y = x ^ { 4 } + 32 x\).
2. Determine whether this stationary point is a maximum or a minimum.
3. For what values of \(x\) does \(x ^ { 4 } + 32 x\) increase as \(x\) increases?

1. A lorry is driven between London and Newcastle.

In a simple model, the cost of the journey \(\pounds C\) when the lorry is driven at a steady speed of \(v\) kilometres per hour is $$C = \frac { 1500 } { v } + \frac { 2 v } { 11 } + 60$$

1. Find, according to this model,
   1. the value of \(v\) that minimises the cost of the journey,
   2. the minimum cost of the journey.
      (Solutions based entirely on graphical or numerical methods are not acceptable.)
2. Prove by using \(\frac { \mathrm { d } ^ { 2 } C } { \mathrm {~d} v ^ { 2 } }\) that the cost is minimised at the speed found in (a)(i).
3. State one limitation of this model.

1. A company makes drinks containers out of metal.

The containers are modelled as closed cylinders with base radius \(r \mathrm {~cm}\) and height \(h \mathrm {~cm}\) and the capacity of each container is \(355 \mathrm {~cm} ^ { 3 }\) The metal used

• for the circular base and the curved side costs 0.04 pence/ \(\mathrm { cm } ^ { 2 }\)
• for the circular top costs 0.09 pence/ \(\mathrm { cm } ^ { 2 }\)

Both metals used are of negligible thickness.

1. Show that the total cost, \(C\) pence, of the metal for one container is given by $$C = 0.13 \pi r ^ { 2 } + \frac { 28.4 } { r }$$
2. Use calculus to find the value of \(r\) for which \(C\) is a minimum, giving your answer to 3 significant figures.
3. Using \(\frac { \mathrm { d } ^ { 2 } C } { \mathrm {~d} r ^ { 2 } }\) prove that the cost is minimised for the value of \(r\) found in part (b).
4. Hence find the minimum value of \(C\), giving your answer to the nearest integer.

1. The curve \(C\) has equation

$$y = \frac { 1 } { 2 } x - \frac { 1 } { 4 } \sin 2 x \quad 0 < x < \pi$$ a. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sin ^ { 2 } x\) b. Find the coordinates of the points of inflection of the curve.

1. A curve \(C\) has equation

$$y = x ^ { 2 } - 2 x - 24 \sqrt { x } , \quad x > 0$$

1. Find (i) \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) (ii) \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\)
2. Verify that \(C\) has a stationary point when \(x = 4\)
3. Determine the nature of this stationary point, giving a reason for your answer.

15. \begin{figure}[h] \end{figure} A company makes toys for children.
Figure 5 shows the design for a solid toy that looks like a piece of cheese.
The toy is modelled so that

• face \(A B C\) is a sector of a circle with radius \(r \mathrm {~cm}\) and centre \(A\)
• angle \(B A C = 0.8\) radians
• faces \(A B C\) and \(D E F\) are congruent
• edges \(A D , C F\) and \(B E\) are perpendicular to faces \(A B C\) and \(D E F\)
• edges \(A D , C F\) and \(B E\) have length \(h \mathrm {~cm}\)

Given that the volume of the toy is \(240 \mathrm {~cm} ^ { 3 }\)

1. show that the surface area of the toy, \(S \mathrm {~cm} ^ { 2 }\), is given by $$S = 0.8 r ^ { 2 } + \frac { 1680 } { r }$$ making your method clear. Using algebraic differentiation,
2. find the value of \(r\) for which \(S\) has a stationary point.
3. Prove, by further differentiation, that this value of \(r\) gives the minimum surface area of the toy.

1. The curve \(C\) has equation

$$x ^ { 2 } \tan y = 9 \quad 0 < y < \frac { \pi } { 2 }$$

1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { - 18 x } { x ^ { 4 } + 81 }$$
2. Prove that \(C\) has a point of inflection at \(x = \sqrt [ 4 ] { 27 }\)

1. (i) Use proof by exhaustion to show that for \(n \in \mathbb { N } , n \leqslant 4\)

$$( n + 1 ) ^ { 3 } > 3 ^ { n }$$ (ii) Given that \(m ^ { 3 } + 5\) is odd, use proof by contradiction to show, using algebra, that \(m\) is even.

1. The curve \(C\) has equation

$$y = 5 x ^ { 4 } - 24 x ^ { 3 } + 42 x ^ { 2 } - 32 x + 11 \quad x \in \mathbb { R }$$

1. Find
   1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
   2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\)
2. 1. Verify that \(C\) has a stationary point at \(x = 1\)
   2. Show that this stationary point is a point of inflection, giving reasons for your answer.

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).

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.

6 Fig. 6 shows the curve with equation \(y = x ^ { 4 } - 6 x ^ { 2 } + 4 x + 5\). \begin{figure}[h] \end{figure}

7

1. By using the definitions of \(\cosh u\) and \(\sinh u\) in terms of \(\mathrm { e } ^ { u }\) and \(\mathrm { e } ^ { - u }\), show that \(\sinh 2 u \equiv 2 \sinh u \cosh u\). The equation of a curve, \(C\), is \(\mathrm { y } = 16 \cosh \mathrm { x } - \sinh 2 \mathrm { x }\).
2. Show that there is only one solution to the equation \(\frac { d ^ { 2 } y } { d x ^ { 2 } } = 0\) You are now given that \(C\) has exactly one point of inflection.
3. Use your answer to part (b) to determine the exact coordinates of this point of inflection. Give your answer in a logarithmic form where appropriate.

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)

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 \(x = - 3\).

8

1. Given that \(\mathrm { e } ^ { - 2 x } = 3\), find the exact value of \(x\).
2. Use integration by parts to find \(\int x \mathrm { e } ^ { - 2 x } \mathrm {~d} x\).
3. A curve has equation \(y = \mathrm { e } ^ { - 2 x } + 6 x\).
   1. Find the exact values of the coordinates of the stationary point of the curve.
   2. Determine the nature of the stationary point.
   3. The region \(R\) is bounded by the curve \(y = \mathrm { e } ^ { - 2 x } + 6 x\), the \(x\)-axis and the lines \(x = 0\) and \(x = 1\). Find the volume of the solid formed when \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis, giving your answer to three significant figures.

7

1. A curve has equation \(y = x ^ { 2 } \mathrm { e } ^ { - \frac { x } { 4 } }\).
   Show that the curve has exactly two stationary points and find the exact values of their coordinates.
   (7 marks)
2. 1. Use integration by parts twice to find the exact value of \(\int _ { 0 } ^ { 4 } x ^ { 2 } \mathrm { e } ^ { - \frac { x } { 4 } } \mathrm {~d} x\).
   2. The region bounded by the curve \(y = 3 x \mathrm { e } ^ { - \frac { x } { 8 } }\), the \(x\)-axis from 0 to 4 and the line \(x = 4\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis to form a solid. Use your answer to part (b)(i) to find the exact value of the volume of the solid generated.

5

1. A curve has equation \(y = \mathrm { e } ^ { 2 x } - 10 \mathrm { e } ^ { x } + 12 x\).
   1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
      (2 marks)
   2. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
      (1 mark)
2. The points \(P\) and \(Q\) are the stationary points of the curve.
   1. Show that the \(x\)-coordinates of \(P\) and \(Q\) are given by the solutions of the equation $$\mathrm { e } ^ { 2 x } - 5 \mathrm { e } ^ { x } + 6 = 0$$ (1 mark)
   2. By using the substitution \(z = \mathrm { e } ^ { x }\), or otherwise, show that the \(x\)-coordinates of \(P\) and \(Q\) are \(\ln 2\) and \(\ln 3\).
   3. Find the \(y\)-coordinates of \(P\) and \(Q\), giving each of your answers in the form \(m + 12 \ln n\), where \(m\) and \(n\) are integers.
   4. Using the answer to part (a)(ii), determine the nature of each stationary point.

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 \(C\).