1.07n Stationary points: find maxima, minima using derivatives

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.
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9. \begin{figure}[h] \end{figure} Figure 1 is a sketch of the curve \(C\) with equation $$y = 2 x ^ { \frac { 3 } { 2 } } ( 4 - x ) \quad x \geqslant 0$$ The point \(P\) is the stationary point of \(C\).

1. Find, using calculus, the \(x\) coordinate of \(P\). The region \(R _ { 1 }\), shown shaded in Figure 1, is bounded by \(C\) and the \(x\)-axis.
   The region \(R _ { 2 }\), also shown shaded in Figure 1, is bounded by \(C\), the \(x\)-axis and the line with equation \(x = k\), where \(k\) is a constant. Given that the area of \(R _ { 1 }\) is equal to the area of \(R _ { 2 }\)
2. find, using calculus, the exact value of \(k\).

10. The curve \(C\) has equation $$y = a x ^ { 3 } - 3 x ^ { 2 } + 3 x + b$$ where \(a\) and \(b\) are constants. Given that

• the point \(( 2,5 )\) lies on \(C\)
• the gradient of the curve at \(( 2,5 )\) is 7
  1. find the value of \(a\) and the value of \(b\).
  2. Prove that \(C\) has no turning points.
     

3. $$f ( x ) = a x ^ { 3 } - x ^ { 2 } + b x + 4$$ where \(a\) and \(b\) are constants. When \(\mathrm { f } ( x )\) is divided by ( \(x + 4\) ), the remainder is - 108

1. Use the remainder theorem to show that $$16 a + b = 24$$ Given also that ( \(2 x - 1\) ) is a factor of \(\mathrm { f } ( x )\),
2. find the value of \(a\) and the value of \(b\).
3. Find \(\mathrm { f } ^ { \prime } ( x )\).
4. Hence find the exact coordinates of the stationary points of the curve with equation \(y = \mathrm { f } ( x )\).

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8. 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 2 shows a sketch of part of the curve \(C\) with equation $$y = \frac { 4 } { 3 } x ^ { 3 } - 11 x ^ { 2 } + k x \quad \text { where } k \text { is a constant }$$ The point \(M\) is the maximum turning point of \(C\) and is shown in Figure 2.
Given that the \(x\) coordinate of \(M\) is 2

1. show that \(k = 28\)
2. Determine the range of values of \(x\) for which \(y\) is increasing. The line \(l\) passes through \(M\) and is parallel to the \(x\)-axis.
   The region \(R\), shown shaded in Figure 2, is bounded by the curve \(C\), the line \(l\) and the \(y\)-axis.
3. Find, by algebraic integration, the exact area of \(R\).

1. The curve \(C\) has equation

$$y = \frac { 12 x ^ { 3 } ( x - 7 ) + 14 x ( 13 x - 15 ) } { 21 \sqrt { x } } \quad x > 0$$

1. Write the equation of \(C\) in the form $$y = a x ^ { \frac { 7 } { 2 } } + b x ^ { \frac { 5 } { 2 } } + c x ^ { \frac { 3 } { 2 } } + d x ^ { \frac { 1 } { 2 } }$$ where \(a , b , c\) and \(d\) are fully simplified constants. The curve \(C\) has three turning points.
   Using calculus,
2. show that the \(x\) coordinates of the three turning points satisfy the equation $$2 x ^ { 3 } - 10 x ^ { 2 } + 13 x - 5 = 0$$ Given that the \(x\) coordinate of one of the turning points is 1
3. find, using algebra, the exact \(x\) coordinates of the other two turning points.
   (Solutions based entirely on calculator technology are not acceptable.)

1. In this question you must show detailed reasoning.

Solutions relying entirely on calculator technology are not acceptable. \begin{figure}[h] \end{figure} Figure 3 shows

• the curve \(C _ { 1 }\) with equation \(y = x ^ { 3 } - 5 x ^ { 2 } + 3 x + 14\)
• the circle \(C _ { 2 }\) with centre \(T\)

The point \(T\) is the minimum turning point of \(C _ { 1 }\) Using Figure 3 and calculus,

1. find the coordinates of \(T\) The curve \(C _ { 1 }\) intersects the circle \(C _ { 2 }\) at the point \(A\) with \(x\) coordinate 2
2. Find an equation of the circle \(C _ { 2 }\) The line \(l\) shown in Figure 3, is the tangent to circle \(C _ { 2 }\) at \(A\)
3. Show that an equation of \(l\) is $$y = \frac { 1 } { 3 } x + \frac { 22 } { 3 }$$ The region \(R\), shown shaded in Figure 3, is bounded by \(C _ { 1 } , l\) and the \(y\)-axis.
4. Find the exact area of \(R\).

9. \begin{figure}[h] \end{figure} In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Figure 3 shows a sketch of part of the curve \(C\) with equation $$y = \frac { 2 } { 3 } x ^ { 2 } - 9 \sqrt { x } + 13 \quad x \geqslant 0$$

1. Find, using calculus, the range of values of \(x\) for which \(y\) is increasing. The point \(P\) lies on \(C\) and has coordinates (9, 40).
   The line \(l\) is the tangent to \(C\) at the point \(P\).
   The finite region \(R\), shown shaded in Figure 3, is bounded by the curve \(C\), the line \(l\), the \(x\)-axis and the \(y\)-axis.
2. Find, using calculus, the exact area of \(R\).

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 \(C\).

1. A diesel lorry is driven from Birmingham to Bury at a steady speed of v kilometres per hour. The total cost of the journey, \(\pounds C\), is given by

$$C = \frac { 1400 } { v } + \frac { 2 v } { 7 } .$$

1. Find the value of \(v\) for which \(C\) is a minimum.
2. Find \(\frac { \mathrm { d } ^ { 2 } C } { \mathrm {~d} v ^ { 2 } }\) and hence verify that \(C\) is a minimum for this value of \(v\).
3. Calculate the minimum total cost of the journey.

10. A solid right circular cylinder has radius \(r \mathrm {~cm}\) and height \(h \mathrm {~cm}\). The total surface area of the cylinder is \(800 \mathrm {~cm} ^ { 2 }\).

1. Show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the cylinder is given by $$V = 400 r - \pi r ^ { 3 }$$ Given that \(r\) varies,
2. use calculus to find the maximum value of \(V\), to the nearest \(\mathrm { cm } ^ { 3 }\).
3. Justify that the value of \(V\) you have found is a maximum. \includegraphics[max width=\textwidth, alt={}, center]{12e54724-64a3-4dc0-b7d5-6ef6cc04124c-16_103_63_2477_1873}

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.

8. \begin{figure}[h] \end{figure} Figure 3 shows a flowerbed. Its shape is a quarter of a circle of radius \(x\) metres with two equal rectangles attached to it along its radii. Each rectangle has length equal to \(x\) metres and width equal to \(y\) metres. Given that the area of the flowerbed is \(4 \mathrm {~m} ^ { 2 }\),

1. show that $$y = \frac { 16 - \pi x ^ { 2 } } { 8 x }$$
2. Hence show that the perimeter \(P\) metres of the flowerbed is given by the equation $$P = \frac { 8 } { x } + 2 x$$
3. Use calculus to find the minimum value of \(P\).
4. Find the width of each rectangle when the perimeter is a minimum. Give your answer to the nearest centimetre.

8. The curve \(C\) has equation \(y = 6 - 3 x - \frac { 4 } { x ^ { 3 } } , x \neq 0\)

1. Use calculus to show that the curve has a turning point \(P\) when \(x = \sqrt { } 2\)
2. Find the \(x\)-coordinate of the other turning point \(Q\) on the curve.
3. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
4. Hence or otherwise, state with justification, the nature of each of these turning points \(P\) and \(Q\).

3. The curve \(C\) has equation $$y = 2 \sqrt { } x + \frac { 18 } { \sqrt { } x } - 1 , \quad x > 0$$

1. Find
   1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
   2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\)
2. Use calculus to find the coordinates of the stationary point of \(C\).
3. Determine whether the stationary point is a maximum or minimum, giving a reason for your answer. \includegraphics[max width=\textwidth, alt={}, center]{e7043e7a-2c8f-425a-8471-f647828cc297-09_138_154_2597_1804}

Find the coordinates of the stationary point on the curve with equation \(y = 2 x ^ { 2 } - 12 x\).

10. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of part of the curve with equation \(y = x ^ { 3 } - 8 x ^ { 2 } + 20 x\). The curve has stationary points \(A\) and \(B\).

1. Use calculus to find the \(x\)-coordinates of \(A\) and \(B\).
2. Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at \(A\), and hence verify that \(A\) is a maximum. The line through \(B\) parallel to the \(y\)-axis meets the \(x\)-axis at the point \(N\).
   The region \(R\), shown shaded in Figure 3, is bounded by the curve, the \(x\)-axis and the line from \(A\) to \(N\).
3. Find \(\int \left( x ^ { 3 } - 8 x ^ { 2 } + 20 x \right) \mathrm { d } x\).
4. Hence calculate the exact area of \(R\).

10. \begin{figure}[h] \end{figure} Figure 4 shows a solid brick in the shape of a cuboid measuring \(2 x \mathrm {~cm}\) by \(x \mathrm {~cm}\) by \(y \mathrm {~cm}\). The total surface area of the brick is \(600 \mathrm {~cm} ^ { 2 }\).

1. Show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the brick is given by $$V = 200 x - \frac { 4 x ^ { 3 } } { 3 }$$ Given that \(x\) can vary,
2. use calculus to find the maximum value of \(V\), giving your answer to the nearest \(\mathrm { cm } ^ { 3 }\).
3. Justify that the value of \(V\) you have found is a maximum.

8. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve with equation \(y = 10 + 8 x + x ^ { 2 } - x ^ { 3 }\).
The curve has a maximum turning point \(A\).

1. Using calculus, show that the \(x\)-coordinate of \(A\) is 2 . The region \(R\), shown shaded in Figure 2, is bounded by the curve, the \(y\)-axis and the line from \(O\) to \(A\), where \(O\) is the origin.
2. Using calculus, find the exact area of \(R\).

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} Figure 2 shows a sketch of part of the curve \(C\) with equation $$y = x ^ { 3 } - 10 x ^ { 2 } + k x$$ where \(k\) is a constant. The point \(P\) on \(C\) is the maximum turning point.
Given that the \(x\)-coordinate of \(P\) is 2 ,

1. show that \(k = 28\). The line through \(P\) parallel to the \(x\)-axis cuts the \(y\)-axis at the point \(N\). The region \(R\) is bounded by \(C\), the \(y\)-axis and \(P N\), as shown shaded in Figure 2.
2. Use calculus to find the exact area of \(R\).

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.