1.07p Points of inflection: using second derivative

9. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of a square based, open top box.
The height of the box is \(h \mathrm {~cm}\), and the base edges each have length \(l \mathrm {~cm}\).
Given that the volume of the box is \(250000 \mathrm {~cm} ^ { 3 }\)

1. show that the external surface area, \(S \mathrm {~cm} ^ { 2 }\), of the box is given by $$S = \frac { 250000 } { h } + 2000 \sqrt { h }$$
2. Use algebraic differentiation to show that \(S\) has a stationary point when \(h = 250 ^ { k }\) where \(k\) is a rational constant to be found.
3. Justify by further differentiation that this value of \(h\) gives the minimum external surface area of the box.
   \includegraphics[max width=\textwidth, alt={}]{515f245f-9c5b-4263-ab2c-0a4f96f3bff0-32_2647_1838_118_116}

1. The height of a river above a fixed point on the riverbed was monitored over a 7-day period.

The height of the river, \(H\) metres, \(t\) days after monitoring began, was given by $$H = \frac { \sqrt { t } } { 20 } \left( 20 + 6 t - t ^ { 2 } \right) + 17 \quad 0 \leqslant t \leqslant 7$$ Given that \(H\) has a stationary value at \(t = \alpha\)

1. use calculus to show that \(\alpha\) satisfies the equation $$5 \alpha ^ { 2 } - 18 \alpha - 20 = 0$$
2. Hence find the value of \(\alpha\), giving your answer to 3 decimal places.
3. Use further calculus to prove that \(H\) is a maximum at this value of \(\alpha\).

9. \begin{figure}[h] \end{figure} Figure 3 shows the plan of a stage in the shape of a rectangle joined to a semicircle. The length of the rectangular part is \(2 x\) metres and the width is \(y\) metres. The diameter of the semicircular part is \(2 x\) metres. The perimeter of the stage is 80 m .

1. Show that the area, \(A \mathrm {~m} ^ { 2 }\), of the stage is given by $$A = 80 x - \left( 2 + \frac { \pi } { 2 } \right) x ^ { 2 } .$$
2. Use calculus to find the value of \(x\) at which \(A\) has a stationary value.
3. Prove that the value of \(x\) you found in part (b) gives the maximum value of \(A\).
4. Calculate, to the nearest \(\mathrm { m } ^ { 2 }\), the maximum area of the stage.

7. The curve \(C\) has equation $$y = 2 x ^ { 3 } - 5 x ^ { 2 } - 4 x + 2$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Using the result from part (a), find the coordinates of the turning points of \(C\).
3. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
4. Hence, or otherwise, determine the nature of the turning points of \(C\).

9. The curve \(C\) has equation \(y = 12 \sqrt { } ( x ) - x ^ { \frac { 3 } { 2 } } - 10 , \quad x > 0\)

1. Use calculus to find the coordinates of the turning point on \(C\).
2. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
3. State the nature of the turning point.

1. The volume \(V \mathrm {~cm} ^ { 3 }\) of a box, of height \(x \mathrm {~cm}\), is given by

$$V = 4 x ( 5 - x ) ^ { 2 } , \quad 0 < x < 5$$

1. Find \(\frac { \mathrm { d } V } { \mathrm {~d} x }\).
2. Hence find the maximum volume of the box.
3. Use calculus to justify that the volume that you found in part (b) is a maximum.

9. \begin{figure}[h] \end{figure} Figure 2 shows a closed box used by a shop for packing pieces of cake. The box is a right prism of height \(h \mathrm {~cm}\). The cross section is a sector of a circle. The sector has radius \(r \mathrm {~cm}\) and angle 1 radian. The volume of the box is \(300 \mathrm {~cm} ^ { 3 }\).

1. Show that the surface area of the box, \(S \mathrm {~cm} ^ { 2 }\), is given by $$S = r ^ { 2 } + \frac { 1800 } { r }$$
2. Use calculus to find the value of \(r\) for which \(S\) is stationary.
3. Prove that this value of \(r\) gives a minimum value of \(S\).
4. Find, to the nearest \(\mathrm { cm } ^ { 2 }\), this minimum value of \(S\).

8. \begin{figure}[h] \end{figure} A cuboid has a rectangular cross-section where the length of the rectangle is equal to twice its width, \(x \mathrm {~cm}\), as shown in Figure 2.
The volume of the cuboid is 81 cubic centimetres.

1. Show that the total length, \(L \mathrm {~cm}\), of the twelve edges of the cuboid is given by $$L = 12 x + \frac { 162 } { x ^ { 2 } }$$
2. Use calculus to find the minimum value of \(L\).
3. Justify, by further differentiation, that the value of \(L\) that you have found is a minimum.

1. The curve with equation

$$y = x ^ { 2 } - 32 \sqrt { } ( x ) + 20 , \quad x > 0$$ has a stationary point \(P\). Use calculus

1. to find the coordinates of \(P\),
2. to determine the nature of the stationary point \(P\).

9. \begin{figure}[h] \end{figure} Figure 4 shows the plan of a pool. The shape of the pool \(A B C D E F A\) consists of a rectangle \(B C E F\) joined to an equilateral triangle \(B F A\) and a semi-circle \(C D E\), as shown in Figure 4. Given that \(A B = x\) metres, \(E F = y\) metres, and the area of the pool is \(50 \mathrm {~m} ^ { 2 }\),

1. show that $$y = \frac { 50 } { x } - \frac { x } { 8 } ( \pi + 2 \sqrt { } 3 )$$
2. Hence show that the perimeter, \(P\) metres, of the pool is given by $$P = \frac { 100 } { x } + \frac { x } { 4 } ( \pi + 8 - 2 \sqrt { } 3 )$$
3. Use calculus to find the minimum value of \(P\), giving your answer to 3 significant figures.
4. Justify, by further differentiation, that the value of \(P\) that you have found is a minimum.

10. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} Figure 4 shows a closed letter box \(A B F E H G C D\), which is made to be attached to a wall of a house. The letter box is a right prism of length \(y \mathrm {~cm}\) as shown in Figure 4. The base \(A B F E\) of the prism is a rectangle. The total surface area of the six faces of the prism is \(S \mathrm {~cm} ^ { 2 }\). The cross section \(A B C D\) of the letter box is a trapezium with edges of lengths \(D A = 9 x \mathrm {~cm}\), \(A B = 4 x \mathrm {~cm} , B C = 6 x \mathrm {~cm}\) and \(C D = 5 x \mathrm {~cm}\) as shown in Figure 5.
The angle \(D A B = 90 ^ { \circ }\) and the angle \(A B C = 90 ^ { \circ }\). The volume of the letter box is \(9600 \mathrm {~cm} ^ { 3 }\).

1. Show that $$y = \frac { 320 } { x ^ { 2 } }$$
2. Hence show that the surface area of the letter box, \(S \mathrm {~cm} ^ { 2 }\), is given by $$S = 60 x ^ { 2 } + \frac { 7680 } { x }$$
3. Use calculus to find the minimum value of \(S\).
4. Justify, by further differentiation, that the value of \(S\) you have found is a minimum.

9. A solid glass cylinder, which is used in an expensive laser amplifier, has a volume of \(75 \pi \mathrm {~cm} ^ { 3 }\).
The cost of polishing the surface area of this glass cylinder is \(\pounds 2\) per \(\mathrm { cm } ^ { 2 }\) for the curved surface area and \(\pounds 3\) per \(\mathrm { cm } ^ { 2 }\) for the circular top and base areas. Given that the radius of the cylinder is \(r \mathrm {~cm}\),

1. show that the cost of the polishing, \(\pounds C\), is given by $$C = 6 \pi r ^ { 2 } + \frac { 300 \pi } { r }$$
2. Use calculus to find the minimum cost of the polishing, giving your answer to the nearest pound.
3. Justify that the answer that you have obtained in part (b) is a minimum.

9. \begin{figure}[h] \end{figure} Figure 4 shows a plan view of a sheep enclosure.
The enclosure \(A B C D E A\), as shown in Figure 4, consists of a rectangle \(B C D E\) joined to an equilateral triangle \(B F A\) and a sector \(F E A\) of a circle with radius \(x\) metres and centre \(F\). The points \(B , F\) and \(E\) lie on a straight line with \(F E = x\) metres and \(10 \leqslant x \leqslant 25\)

1. Find, in \(\mathrm { m } ^ { 2 }\), the exact area of the sector \(F E A\), giving your answer in terms of \(x\), in its simplest form. Given that \(B C = y\) metres, where \(y > 0\), and the area of the enclosure is \(1000 \mathrm {~m} ^ { 2 }\),
2. show that $$y = \frac { 500 } { x } - \frac { x } { 24 } ( 4 \pi + 3 \sqrt { 3 } )$$
3. Hence show that the perimeter \(P\) metres of the enclosure is given by $$P = \frac { 1000 } { x } + \frac { x } { 12 } ( 4 \pi + 36 - 3 \sqrt { 3 } )$$
4. Use calculus to find the minimum value of \(P\), giving your answer to the nearest metre.
5. Justify, by further differentiation, that the value of \(P\) you have found is a minimum.

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 \(C\).

6

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

10. \includegraphics[max width=\textwidth, alt={}, center]{6ef55dbd-f18d-4264-b80c-d181473ca7b3-3_531_786_246_523} The diagram shows an open-topped cylindrical container made from cardboard. The cylinder is of height \(h \mathrm {~cm}\) and base radius \(r \mathrm {~cm}\). Given that the area of card used to make the container is \(192 \pi \mathrm {~cm} ^ { 2 }\),

1. show that the capacity of the container, \(\mathrm { V } \mathrm { cm } ^ { 3 }\), is given by $$V = 96 \pi r - \frac { 1 } { 2 } \pi r ^ { 3 } .$$
2. Find the value of \(r\) for which \(V\) is stationary.
3. Find the corresponding value of \(V\) in terms of \(\pi\).
4. Determine whether this is a maximum or a minimum value of \(V\).

10 A curve has equation \(y = x ^ { 3 } - 6 x ^ { 2 } + 12\).

1. Use calculus to find the coordinates of the turning points of this curve. Determine also the nature of these turning points.
2. Find, in the form \(y = m x + c\), the equation of the normal to the curve at the point \(( 2 , - 4 )\).

6 A curve has gradient given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { 2 } - 6 x + 9\). Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
Show that the curve has a stationary point of inflection when \(x = 3\).

10 Fig. 10 shows the curve with equation \(y = x ^ { 2 } + \frac { 16 } { x }\). \begin{figure}[h] \end{figure}

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Hence calculate the coordinates of the stationary point on the curve.
3. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and explain why this confirms that he stationary point is a minimum.
4. Using the trapezium rule with 4 intervals, estimate the area between the curve and the \(x\) axis between \(x = 2\) and \(x = 4\).
5. State, giving a reason, whether this estimate of the area under-estimates or over-estimates the true area beneath the curve.

2 A cubic curve has equation \(y = x ^ { 3 } - 3 x ^ { 2 } + 1\).

1. Use calculus to find the coordinates of the turning points on this curve. Determine the nature of these turning points.
2. Show that the tangent to the curve at the point where \(x = - 1\) has gradient 9 . Find the coordinates of the other point, P , on the curve at which the tangent has gradient 9 and find the equation of the normal to the curve at P . Show that the area of the triangle bounded by the normal at P and the \(x\) - and \(y\)-axes is 8 square units.

3 A curve has equation \(y = x + \frac { 1 } { x }\).
Use calculus to show that the curve has a turning point at \(x = 1\).
Show also that this point is a minimum.

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

1. Show that the curve meets the \(x\)-axis at the origin and at \(x = \pm a\), stating the value of \(a\).
2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\). Hence show that the origin is a minimum point on the curve. Find the \(x\)-coordinates of the maximum points.
3. Use calculus to find the area of the region bounded by the curve and the \(x\)-axis between \(x = 0\) and \(x = a\), using the value you found for \(a\) in part (i).

2 A curve has equation \(y = x ^ { 3 } - 6 x ^ { 2 } + 12\).

1. Use calculus to find the coordinates of the turning points of this curve. Determine also the nature of these turning points.
2. Find, in the form \(y = m x + c\), the equation of the normal to the curve at the point \(( 2 , - 4 )\).

7 A curve has equation \(y = ( x + 2 ) \left( x ^ { 2 } - 3 x + 5 \right)\).

1. Find the coordinates of the minimum point, justifying that it is a minimum.
2. Calculate the discriminant of \(x ^ { 2 } - 3 x + 5\).
3. Explain why \(( x + 2 ) \left( x ^ { 2 } - 3 x + 5 \right)\) is always positive for \(x > - 2\).

8

1. Find the coordinates of the stationary point on the curve \(y = 3 x ^ { 2 } - \frac { 6 } { x } - 2\).
2. Determine whether the stationary point is a maximum point or a minimum point.