Stationary points and optimisation

3 Find the set of values of \(x\) for which \(x ^ { 2 } - 7 x\) is a decreasing function.

Given \(y = \frac { x ^ { 3 } } { 3 } - 2 x ^ { 2 } + 3 x + 5\)

1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), simplifying each term.
2. Hence find the set of values of \(x\) for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } > 0\)

The diagram below shows a sketch of the graph of \(y = f'(x)\) for the interval \([x_1, x_5]\). \includegraphics{figure_13}

1. Find the interval on which \(f(x)\) is both decreasing and convex. Give reasons for your answer. [2]
2. Write down the \(x\)-coordinate of a point of inflection of the graph of \(y = f(x)\). [1]

Find the coordinates of the stationary point on the curve with equation \(y = 2x^2 - 12x\). [4]

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

11. \begin{figure}[h] \end{figure} Figure 5 shows a sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\).
The curve \(C\) crosses the \(x\)-axis at the origin, \(O\), and at the points \(A\) and \(B\) as shown in Figure 5. Given that $$f ^ { \prime } ( x ) = k - 4 x - 3 x ^ { 2 }$$ where \(k\) is constant,

1. show that \(C\) has a point of inflection at \(x = - \frac { 2 } { 3 }\) Given also that the distance \(A B = 4 \sqrt { 2 }\)
2. find, showing your working, the integer value of \(k\).

4 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.

2. A curve has equation $$y = x ^ { 3 } - x ^ { 2 } - 16 x + 2$$

1. Using calculus, find the \(x\) coordinates of the stationary points of the curve.
2. Justify, by further calculus, the nature of all of the stationary points of 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.

The function f is defined by $$f(x) = x^2 + 2 \cos x \text{ for } -\pi \leq x \leq \pi$$ Determine whether the curve with equation \(y = f(x)\) has a point of inflection at the point where \(x = 0\) Fully justify your answer. [4 marks]

10

1. Given that \(y = \frac { 1 } { 3 } x ^ { 3 } - 9 x\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Find the coordinates of the stationary points on the curve \(y = \frac { 1 } { 3 } x ^ { 3 } - 9 x\).
3. Determine whether each stationary point is a maximum point or a minimum point.
4. Given that \(24 x + 3 y + 2 = 0\) is the equation of the tangent to the curve at the point ( \(p , q\) ), find \(p\) and \(q\).

2 A curve has equation \(y = \mathrm { f } ( x )\) The curve has a point of inflection at \(x = 7\) It is given that \(\mathrm { f } ^ { \prime } ( 7 ) = a\) and \(\mathrm { f } ^ { \prime \prime } ( 7 ) = b\), where \(a\) and \(b\) are real numbers. Identify which one of the statements below must be true.
Circle your answer. \(\mathrm { f } ^ { \prime } ( 7 ) \neq 0\) \(\mathrm { f } ^ { \prime } ( 7 ) = 0\) \(\mathrm { f } ^ { \prime \prime } ( 7 ) \neq 0\) \(\mathrm { f } ^ { \prime \prime } ( 7 ) = 0\)

3 A cylindrical metal tin of radius \(r \mathrm {~cm}\) is closed at both ends. It has a volume of \(16000 \pi \mathrm {~cm} ^ { 3 }\).

1. Show that its total surface area, \(A \mathrm {~cm} ^ { 2 }\), is given by \(A = 2 \pi r ^ { 2 } + 32000 \pi r ^ { - 1 }\).
2. Use calculus to determine the minimum total surface area of the tin. You should justify that it is a minimum.

1. In this question you must show all stages of your working.

\section*{Solutions based entirely on calculator technology are not acceptable.} \begin{figure}[h] \end{figure} A brick is in the shape of a cuboid with width \(x \mathrm {~cm}\) ,length \(3 x \mathrm {~cm}\) and height \(h \mathrm {~cm}\) ,as shown in Figure 2. The volume of the brick is \(972 \mathrm {~cm} ^ { 3 }\)

1. Show that the surface area of the brick,\(S \mathrm {~cm} ^ { 2 }\) ,is given by $$S = 6 x ^ { 2 } + \frac { 2592 } { x }$$
2. Find \(\frac { \mathrm { d } S } { \mathrm {~d} x }\)
3. Hence find the value of \(x\) for which \(S\) is stationary.
4. Find \(\frac { \mathrm { d } ^ { 2 } S } { \mathrm {~d} x ^ { 2 } }\) and hence show that the value of \(x\) found in part(c)gives the minimum value of \(S\) .
5. Hence find the minimum surface area of the brick.

7 \includegraphics[max width=\textwidth, alt={}, center]{7fc02f90-8f8b-4153-bba1-dc0807124e96-5_421_944_251_242} The diagram shows a model for the roof of a toy building. The roof is in the form of a solid triangular prism \(A B C D E F\). The base \(A C F D\) of the roof is a horizontal rectangle, and the crosssection \(A B C\) of the roof is an isosceles triangle with \(A B = B C\). The lengths of \(A C\) and \(C F\) are \(2 x \mathrm {~cm}\) and \(y \mathrm {~cm}\) respectively, and the height of \(B E\) above the base of the roof is \(x \mathrm {~cm}\). The total surface area of the five faces of the roof is \(600 \mathrm {~cm} ^ { 2 }\) and the volume of the roof is \(V \mathrm {~cm} ^ { 3 }\).

1. Show that \(V = k x \left( 300 - x ^ { 2 } \right)\), where \(k = \sqrt { a } + b\) and \(a\) and \(b\) are integers to be determined.
2. Use differentiation to determine the value of \(x\) for which the volume of the roof is a maximum.
3. Find the maximum volume of the roof. Give your answer in \(\mathrm { cm } ^ { 3 }\), correct to the nearest integer.
4. Explain why, for this roof, \(x\) must be less than a certain value, which you should state.

A curve is such that \(\frac{\text{d}y}{\text{d}x} = 3x^2 + ax + b\). The curve has stationary points at \((-1, 2)\) and \((3, k)\). Find the values of the constants \(a\), \(b\) and \(k\). [8]

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).

\includegraphics{figure_5} Figure 5 shows a sketch of the curve \(C\) with equation \(y = f(x)\). The curve \(C\) crosses the \(x\)-axis at the origin, \(O\), and at the points \(A\) and \(B\) as shown in Figure 5. Given that $$f'(x) = k - 4x - 3x^2$$ where \(k\) is constant.

1. show that \(C\) has a point of inflection at \(x = -\frac{2}{3}\) [3] Given also that the distance \(AB = 4\sqrt{2}\)
2. find, showing your working, the integer value of \(k\). [5]

4

1. Differentiate \(x ^ { 3 } - 3 x ^ { 2 } - 9 x\). Hence find the \(x\)-coordinates of the stationary points on the curve \(y = x ^ { 3 } - 3 x ^ { 2 } - 9 x\), showing which is the maximum and which the minimum.
2. Find, in exact form, the coordinates of the points at which the curve crosses the \(x\)-axis.
3. Sketch the curve.

10 The equation of a curve is \(y = 7 + 6 x - x ^ { 2 }\).

1. Use calculus to find the coordinates of the turning point on this curve. Find also the coordinates of the points of intersection of this curve with the axes, and sketch the curve.
2. Find \(\int _ { 1 } ^ { 5 } \left( 7 + 6 x - x ^ { 2 } \right) \mathrm { d } x\), showing your working.
3. The curve and the line \(y = 12\) intersect at ( 1,12 ) and ( 5,12 ). Using your answer to part (ii), find the area of the finite region between the curve and the line \(y = 12\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{15b8f97b-c058-409f-907f-cb0a6102abc4-5_643_1034_331_513} \captionsetup{labelformat=empty} \caption{Fig. 11}
   \end{figure} The equation of the curve shown in Fig. 11 is \(y = x ^ { 3 } - 6 x + 2\).

The curve \(y = ax^4 + bx^3 + 18x^2\) has a point of inflection at \((1, 11)\).

1. Show that \(2a + b + 6 = 0\). [2]
2. Find the values of the constants \(a\) and \(b\) and show that the curve has another point of inflection at \((3, 27)\). [8]
3. Sketch the curve, identifying all the stationary points including their nature. [6]

A rectangle is inscribed in a semicircle with centre \(O\) and radius 4. The point \(P ( x , y )\) is the vertex of the rectangle in the first quadrant as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{72bb1603-edbd-4e2e-bf2b-f33bb667e61b-6_553_929_1430_593}
a) Express the area \(A\) of the rectangle as a function of \(x\).
b) Show that the maximum value of \(A\) occurs when \(y = x\).

5 \includegraphics[max width=\textwidth, alt={}, center]{581ef815-59f0-434e-a7ec-9128e74c0323-2_256_1113_1366_516} The diagram shows a rectangular enclosure, with a wall forming one side. A rope, of length 20 metres, is used to form the remaining three sides. The width of the enclosure is x metres.

1. Show that the enclosed area, \(\mathrm { Am } ^ { 2 }\), is given by $$A = 20 x - 2 x ^ { 2 } .$$
2. Use differentiation to find the maximum value of A .

1. Find an expression, in terms of \(x\), for the area \(A\) of \(R\).
2. Show that \(\frac { \mathrm { d } A } { \mathrm {~d} x } = \frac { 1 } { 4 } ( \pi - 2 x - 2 \sin x ) \sec ^ { 2 } \frac { x } { 2 }\).
3. Prove that the maximum value of \(A\) occurs when \(\frac { \pi } { 4 } < x < \frac { \pi } { 3 }\).
4. Prove that \(\tan \frac { \pi } { 8 } = \sqrt { } 2 - 1\).
5. Show that the maximum value of \(A > \frac { \pi } { 4 } ( \sqrt { } 2 - 1 )\).

1. Express \(h\) in terms of \(r\) and hence show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the cylinder is given by $$V = 12 \pi r ^ { 2 } - 2 \pi r ^ { 3 }$$
2. Given that \(r\) varies, find the stationary value of \(V\).

8 \includegraphics[max width=\textwidth, alt={}, center]{e3942549-bfc0-432a-bf49-7d01d44af01a-6_533_524_246_772} The diagram shows a container which consists of a cylinder with a solid base and a hemispherical top. The radius of the cylinder is \(r \mathrm {~cm}\) and the height is \(h \mathrm {~cm}\). The container is to be made of thin plastic. The volume of the container is \(45 \pi \mathrm {~cm} ^ { 3 }\).

1. Show that the surface area of the container, \(A \mathrm {~cm} ^ { 2 }\), is given by $$A = \frac { 5 } { 3 } \pi r ^ { 2 } + \frac { 90 \pi } { r } .$$ [The volume of a sphere is \(V = \frac { 4 } { 3 } \pi r ^ { 3 }\) and the surface area of a sphere is \(S = 4 \pi r ^ { 2 }\).]
2. Use calculus to find the minimum surface area of the container, justifying that it is a minimum.
3. Suggest a reason why the manufacturer would wish to minimise the surface area.

Prove that the curve with equation $$y = 2x^5 + 5x^4 + 10x^3 - 8$$ has only one stationary point, stating its coordinates. [6 marks]

9. The total cost \(C\), in \(\pounds\), for a certain car journey, is modelled by $$C = \frac { 200 } { V } + \frac { 2 V } { 25 } , V > 30 ,$$ where \(V\) is the average speed in miles per hour.
a) Find the value of \(V\) for which \(C\) is stationary.

3 A surface has equation \(z = x ^ { 2 } y ^ { 2 } - 3 x y + 2 x + y\) for all real values of \(x\) and \(y\). Determine the coordinates of all stationary points of this surface.

2. $$f ( x ) = 2 - x - x ^ { 3 } .$$ Show that \(\mathrm { f } ( x )\) is decreasing for all values of \(x\).

\includegraphics{figure_2} A rectangular sheet of metal measures 50 cm by 40 cm. Squares of side \(x\) cm are cut from each corner of the sheet and the remainder is folded along the dotted lines to make an open tray, as shown in Fig. 2.

1. Show that the volume, \(V\) cm\(^3\), of the tray is given by $$V = 4x(x^2 - 45x + 500)$$ [3]
2. State the range of possible values of \(x\). [1]
3. Find the value of \(x\) for which \(V\) is a maximum. [4]
4. Hence find the maximum value of \(V\). [2]
5. Justify that the value of \(V\) you found in part (d) is a maximum. [2]

2. In this question you must show all stages of your working. \section*{Solutions relying on calculator technology are not acceptable.} A curve has equation $$y = 3 x ^ { 5 } + 4 x ^ { 3 } - x + 5$$ The points \(P\) and \(Q\) lie on the curve.
The gradient of the curve at both point \(P\) and point \(Q\) is 2
Find the \(x\) coordinates of \(P\) and \(Q\).

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.