1.07n Stationary points: find maxima, minima using derivatives

3 A curve has equation \(y = 2 x ^ { 5 } + 5 x ^ { 4 } - 1\).

1. Find:
   1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
   2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\)
2. The point on the curve where \(x = - 1\) is \(P\).
   1. Determine whether \(y\) is increasing or decreasing at \(P\), giving a reason for your answer.
   2. Find an equation of the tangent to the curve at \(P\).
3. The point \(Q ( - 2,15 )\) also lies on the curve. Verify that \(Q\) is a maximum point of the curve.
   [0pt] [4 marks]

6 The diagram shows a cylindrical container of radius \(r \mathrm {~cm}\) and height \(h \mathrm {~cm}\). The container has an open top and a circular base. \includegraphics[max width=\textwidth, alt={}, center]{c7f38f7e-75aa-4b41-96fd-f38f968c225c-12_389_426_404_751} The external surface area of the container's curved surface and base is \(48 \pi \mathrm {~cm} ^ { 2 }\).
When the radius of the base is \(r \mathrm {~cm}\), the volume of the container is \(V \mathrm {~cm} ^ { 3 }\).

1. 1. Find an expression for \(h\) in terms of \(r\).
   2. Show that \(V = 24 \pi r - \frac { \pi } { 2 } r ^ { 3 }\).
2. 1. Find \(\frac { \mathrm { d } V } { \mathrm {~d} r }\).
   2. Find the positive value of \(r\) for which \(V\) is stationary, and determine whether this stationary value is a maximum value or a minimum value.
      [0pt] [4 marks]

8 The gradient, \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), at the point \(( x , y )\) on a curve is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 54 + 27 x - 6 x ^ { 2 }$$

1. 1. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
   2. The curve passes through the point \(P \left( - 1 \frac { 1 } { 2 } , 4 \right)\). Verify that the curve has a minimum point at \(P\).
2. 1. Show that at the points on the curve where \(y\) is decreasing $$2 x ^ { 2 } - 9 x - 18 > 0$$
   2. Solve the inequality \(2 x ^ { 2 } - 9 x - 18 > 0\).

6

1. 1. Using the binomial expansion, or otherwise, express \(( 2 + x ) ^ { 3 }\) in the form \(8 + a x + b x ^ { 2 } + x ^ { 3 }\), where \(a\) and \(b\) are integers. (3 marks)
   2. Write down the expansion of \(( 2 - x ) ^ { 3 }\).
2. Hence show that \(( 2 + x ) ^ { 3 } - ( 2 - x ) ^ { 3 } = 24 x + 2 x ^ { 3 }\).
3. Hence show that the curve with equation $$y = ( 2 + x ) ^ { 3 } - ( 2 - x ) ^ { 3 }$$ has no stationary points.

1 Given that \(y = 16 x + x ^ { - 1 }\), find the two values of \(x\) for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\).
(5 marks)

5 A curve, drawn from the origin \(O\), crosses the \(x\)-axis at the point \(P ( 4,0 )\).
The normal to the curve at \(P\) meets the \(y\)-axis at the point \(Q\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{14c2acbb-5f3e-40e2-8b88-162341ab9f71-3_526_629_916_813} The curve, defined for \(x \geqslant 0\), has equation $$y = 4 x ^ { \frac { 1 } { 2 } } - x ^ { \frac { 3 } { 2 } }$$

1. 1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
      (3 marks)
   2. Show that the gradient of the curve at \(P ( 4,0 )\) is - 2 .
   3. Find an equation of the normal to the curve at \(P ( 4,0 )\).
   4. Find the \(y\)-coordinate of \(Q\) and hence find the area of triangle \(O P Q\).
   5. The curve has a maximum point \(M\). Find the \(x\)-coordinate of \(M\).
2. 1. Find \(\int \left( 4 x ^ { \frac { 1 } { 2 } } - x ^ { \frac { 3 } { 2 } } \right) \mathrm { d } x\).
   2. Find the total area of the region bounded by the curve and the lines \(P Q\) and \(Q O\).

5

1. By using the binomial expansion, or otherwise, express \(( 1 + 2 x ) ^ { 4 }\) in the form $$1 + a x + b x ^ { 2 } + c x ^ { 3 } + 16 x ^ { 4 }$$ where \(a\), \(b\) and \(c\) are integers.
2. Hence show that \(( 1 + 2 x ) ^ { 4 } + ( 1 - 2 x ) ^ { 4 } = 2 + 48 x ^ { 2 } + 32 x ^ { 4 }\).
3. Hence show that the curve with equation $$y = ( 1 + 2 x ) ^ { 4 } + ( 1 - 2 x ) ^ { 4 }$$ has just one stationary point and state its coordinates.

5 A curve has equation \(y = \frac { 1 } { x ^ { 3 } } + 48 x\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Hence find the equation of each of the two tangents to the curve that are parallel to the \(x\)-axis.
3. Find an equation of the normal to the curve at the point \(( 1,49 )\).

5 The point \(P ( 2,8 )\) lies on a curve, and the point \(M\) is the only stationary point of the curve. The curve has equation \(y = 6 + 2 x - \frac { 8 } { x ^ { 2 } }\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Show that the normal to the curve at the point \(P ( 2,8 )\) has equation \(x + 4 y = 34\).
3. 1. Show that the stationary point \(M\) lies on the \(x\)-axis.
   2. Hence write down the equation of the tangent to the curve at \(M\).
4. The tangent to the curve at \(M\) and the normal to the curve at \(P\) intersect at the point \(T\). Find the coordinates of \(T\).

5 The diagram shows part of a curve with a maximum point \(M\). \includegraphics[max width=\textwidth, alt={}, center]{258f0400-6e3b-406c-9f86-acc9fff4e094-4_480_645_354_694} The curve is defined for \(x \geqslant 0\) by the equation $$y = 6 x - 2 x ^ { \frac { 3 } { 2 } }$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   (3 marks)
2. 1. Hence find the coordinates of the maximum point \(M\).
   2. Write down the equation of the normal to the curve at \(M\).
3. The point \(P \left( \frac { 9 } { 4 } , \frac { 27 } { 4 } \right)\) lies on the curve.
   1. Find an equation of the normal to the curve at the point \(P\), giving your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are positive integers.
   2. The normals to the curve at the points \(M\) and \(P\) intersect at the point \(R\). Find the coordinates of \(R\). \(6 \quad\) A curve \(C\), defined for \(0 \leqslant x \leqslant 2 \pi\) by the equation \(y = \sin x\), where \(x\) is in radians, is sketched below. The region bounded by the curve \(C\), the \(x\)-axis from 0 to 2 and the line \(x = 2\) is shaded. \includegraphics[max width=\textwidth, alt={}, center]{258f0400-6e3b-406c-9f86-acc9fff4e094-5_441_789_466_612}

6 A curve has the equation $$y = \frac { 12 + x ^ { 2 } \sqrt { x } } { x } , \quad x > 0$$

1. Express \(\frac { 12 + x ^ { 2 } \sqrt { x } } { x }\) in the form \(12 x ^ { p } + x ^ { q }\).
2. 1. Hence find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   2. Find an equation of the normal to the curve at the point on the curve where \(x = 4\).
   3. The curve has a stationary point \(P\). Show that the \(x\)-coordinate of \(P\) can be written in the form \(2 ^ { k }\), where \(k\) is a rational number.

4 A curve is defined for \(x > 0\). The gradient of the curve at the point \(( x , y )\) is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 } { x ^ { 2 } } - \frac { x } { 4 }$$

1. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
2. The curve has a stationary point \(M\) whose \(y\)-coordinate is \(\frac { 5 } { 2 }\).
   1. Find the \(x\)-coordinate of \(M\).
   2. Use your answers to parts (a) and (b)(i) to show that \(M\) is a maximum point.
   3. Find the equation of the curve.

3 The diagram shows a curve with a maximum point \(M\). \includegraphics[max width=\textwidth, alt={}, center]{e183578a-29a8-4112-b941-06c8894ed078-06_512_867_354_589} The curve is defined for \(x > 0\) by the equation $$y = 6 x ^ { \frac { 1 } { 2 } } - x - 3$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Hence find the \(y\)-coordinate of the maximum point \(M\).
3. Find an equation of the normal to the curve at the point \(P ( 4,5 )\).
4. It is given that the normal to the curve at \(P\), when translated by the vector \(\left[ \begin{array} { l } k \\ 0 \end{array} \right]\), passes through the point \(M\). Find the value of the constant \(k\).
   [0pt] [3 marks]

7. A geometric series is \(a + a r + a r ^ { 2 } + \ldots\)

1. Prove that the sum of the first \(n\) terms of this series is given by \(S _ { n } = \frac { a \left( 1 - r ^ { n } \right) } { 1 - r }\). The second and fourth terms of the series are 3 and 1.08 respectively.
   Given that all terms in the series are positive, find
2. the value of \(r\) and the value of \(a\),
3. the sum to infinity of the series. \includegraphics[max width=\textwidth, alt={}, center]{1033051d-18bf-4734-a556-4c8e1c789992-4_764_1159_294_299} Fig. 2 shows part of the curve with equation \(y = x ^ { 3 } - 6 x ^ { 2 } + 9 x\). The curve touches the \(x\)-axis at \(A\) and has a maximum turning point at \(B\).
   1. Show that the equation of the curve may be written as \(y = x ( x - 3 ) ^ { 2 }\), and hence write down the coordinates of \(A\).
   2. Find the coordinates of \(B\). The shaded region \(R\) is bounded by the curve and the \(x\)-axis.
   3. Find the area of \(R\).

3. Given that \(\mathrm { f } ( x ) = 15 - 7 x - 2 x ^ { 2 }\),

1. find the coordinates of all points at which the graph of \(y = \mathrm { f } ( x )\) crosses the coordinate axes.
2. Sketch the graph of \(y = \mathrm { f } ( x )\).
3. Calculate the coordinates of the stationary point of \(\mathrm { f } ( x )\).
   [0pt] [P1 June 2002 Question 3]

6. A container made from thin metal is in the shape of a right circular cylinder with height \(h \mathrm {~cm}\) and base radius \(r \mathrm {~cm}\). The container has no lid. When full of water, the container holds \(500 \mathrm {~cm} ^ { 3 }\) of water.

1. Show that the exterior surface area, \(A \mathrm {~cm} ^ { 2 }\), of the container is given by \(A = \pi r ^ { 2 } + \frac { 1000 } { r }\).
2. Find the value of \(r\) for which \(A\) is a minimum.
3. Prove that this value of \(r\) gives a minimum value of \(A\).
4. Calculate the minimum value of \(A\), giving your answer to the nearest integer.

9. \begin{figure}[h] \end{figure} Figure 3 shows a design consisting of two rectangles measuring \(x \mathrm {~cm}\) by \(y \mathrm {~cm}\) joined to a circular sector of radius \(x \mathrm {~cm}\) and angle 0.5 radians. Given that the area of the design is \(50 \mathrm {~cm} ^ { 2 }\),

1. show that the perimeter, \(P\) cm, of the design is given by $$P = 2 x + \frac { 100 } { x }$$
2. Find the value of \(x\) for which \(P\) is a minimum.
3. Show that \(P\) is a minimum for this value of \(x\).
4. Find the minimum value of \(P\) in the form \(k \sqrt { 2 }\).

6. \begin{figure}[h] \end{figure} Figure 1 shows the curve with equation \(y = 4 x + \frac { 1 } { x } , x > 0\).

1. Find the coordinates of the minimum point of the curve. The shaded region \(R\) is bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 4\).
2. Use the trapezium rule with three intervals of equal width to estimate the area of \(R\).

8. \begin{figure}[h] \end{figure} Figure 3 shows the curve with equation \(y = \sqrt { x } + \frac { 8 } { x ^ { 2 } } , x > 0\).

1. Find the coordinates of the minimum point of the curve.
2. Show that the area of the shaded region bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 9\) is \(24 \frac { 4 } { 9 }\).

7. $$f ( x ) = 2 + 6 x ^ { 2 } - x ^ { 3 }$$

1. Find the coordinates of the stationary points of the curve \(y = \mathrm { f } ( x )\).
2. Determine whether each stationary point is a maximum or minimum point.
3. Sketch the curve \(y = \mathrm { f } ( x )\).
4. State the set of values of \(k\) for which the equation \(\mathrm { f } ( x ) = k\) has three solutions.

9. \begin{figure}[h] \end{figure} Figure 2 shows a tray made from sheet metal.
The horizontal base is a rectangle measuring \(8 x \mathrm {~cm}\) by \(y \mathrm {~cm}\) and the two vertical sides are trapezia of height \(x \mathrm {~cm}\) with parallel edges of length \(8 x \mathrm {~cm}\) and \(10 x \mathrm {~cm}\). The remaining two sides are rectangles inclined at \(45 ^ { \circ }\) to the horizontal. Given that the capacity of the tray is \(900 \mathrm {~cm} ^ { 3 }\),

1. find an expression for \(y\) in terms of \(x\),
2. show that the area of metal used to make the tray, \(A \mathrm {~cm} ^ { 2 }\), is given by $$A = 18 x ^ { 2 } + \frac { 200 ( 4 + \sqrt { 2 } ) } { x } ,$$
3. find to 3 significant figures, the value of \(x\) for which \(A\) is stationary,
4. find the minimum value of \(A\) and show that it is a minimum.

9. \(f ( x ) = x ^ { 3 } - 4 x ^ { 2 } - 3 x + 18\).

1. Show that \(( x - 3 )\) is a factor of \(\mathrm { f } ( x )\).
2. Fully factorise \(\mathrm { f } ( x )\).
3. Using your answer to part (b), write down the coordinates of one of the turning points of the curve \(y = \mathrm { f } ( x )\) and give a reason for your answer.
4. Using differentiation, find the \(x\)-coordinate of the other turning point of the curve \(y = \mathrm { f } ( x )\).

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.

8

1. Given that \(\mathrm { e } ^ { - 2 x } = 4\), find the exact value of \(x\).
2. The diagram shows the curve \(y = 4 \mathrm { e } ^ { - 2 x } - \mathrm { e } ^ { - 4 x }\). \includegraphics[max width=\textwidth, alt={}, center]{6761e676-48ae-47e9-9617-153342cdf5c4-9_490_1185_463_440} The curve crosses the \(y\)-axis at the point \(A\), the \(x\)-axis at the point \(B\), and has a stationary point at \(M\).
   1. State the \(y\)-coordinate of \(A\).
   2. Find the \(x\)-coordinate of \(B\), giving your answer in an exact form.
   3. Find the \(x\)-coordinate of the stationary point, \(M\), giving your answer in an exact form.
   4. The shaded region \(R\) is bounded by the curve \(y = 4 \mathrm { e } ^ { - 2 x } - \mathrm { e } ^ { - 4 x }\), the lines \(x = 0\) and \(x = \ln 2\) and the \(x\)-axis. Find the volume of the solid generated when the region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis, giving your answer in the form \(\frac { p } { q } \pi\), where \(p\) and \(q\) are integers.
      (7 marks)

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.