1.07n Stationary points: find maxima, minima using derivatives

10 The curve with equation \(y = x ^ { 3 } - 2 x ^ { 2 } + 5 x\) passes through the origin.

1. Show that the curve has no stationary points.
2. Denoting the gradient of the curve by \(m\), find the stationary value of \(m\) and determine its nature.
3. Showing all necessary working, find the area of the region enclosed by the curve, the \(x\)-axis and the line \(x = 6\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

11 \includegraphics[max width=\textwidth, alt={}, center]{58d65166-2b1a-4b58-9859-afe919c0a3a9-18_643_969_258_587} The diagram shows part of the curve \(y = \frac { x } { 2 } + \frac { 6 } { x }\). The line \(y = 4\) intersects the curve at the points \(P\) and \(Q\).

1. Show that the tangents to the curve at \(P\) and \(Q\) meet at a point on the line \(y = x\).
2. Find, showing all necessary working, the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Give your answer in terms of \(\pi\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

11 \includegraphics[max width=\textwidth, alt={}, center]{ea402a1d-3632-4637-9198-2365715b5246-18_645_723_258_573} The diagram shows part of the curve \(y = ( x + 1 ) ^ { 2 } + ( x + 1 ) ^ { - 1 }\) and the line \(x = 1\). The point \(A\) is the minimum point on the curve.

1. Show that the \(x\)-coordinate of \(A\) satisfies the equation \(2 ( x + 1 ) ^ { 3 } = 1\) and find the exact value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at \(A\).
2. Find, showing all necessary working, the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

10 A curve for which \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 2 x - 5\) has a stationary point at \(( 3,6 )\).

1. Find the equation of the curve.
2. Find the \(x\)-coordinate of the other stationary point on the curve.
3. Determine the nature of each of the stationary points. \includegraphics[max width=\textwidth, alt={}, center]{ebf16cae-1e80-44d2-9c51-630f5dc3c11f-20_700_616_262_762} The diagram shows part of the curve \(y = \frac { 3 } { \sqrt { ( 1 + 4 x ) } }\) and a point \(P ( 2,1 )\) lying on the curve. The normal to the curve at \(P\) intersects the \(x\)-axis at \(Q\).
4. Show that the \(x\)-coordinate of \(Q\) is \(\frac { 16 } { 9 }\).
5. Find, showing all necessary working, the area of the shaded region.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

11 \includegraphics[max width=\textwidth, alt={}, center]{ed5b77ae-6eac-4e73-bc43-613433abd3e1-16_723_942_260_598} The diagram shows part of the curve \(y = \sqrt { } ( 4 x + 1 ) + \frac { 9 } { \sqrt { } ( 4 x + 1 ) }\) and the minimum point \(M\).

1. Find expressions for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\int y \mathrm {~d} x\).
2. Find the coordinates of \(M\).
   The shaded region is bounded by the curve, the \(y\)-axis and the line through \(M\) parallel to the \(x\)-axis.
3. Find, showing all necessary working, the area of the shaded region.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

6 A vacuum flask (for keeping drinks hot) is modelled as a closed cylinder in which the internal radius is \(r \mathrm {~cm}\) and the internal height is \(h \mathrm {~cm}\). The volume of the flask is \(1000 \mathrm {~cm} ^ { 3 }\). A flask is most efficient when the total internal surface area, \(A \mathrm {~cm} ^ { 2 }\), is a minimum.

1. Show that \(A = 2 \pi r ^ { 2 } + \frac { 2000 } { r }\).
2. Given that \(r\) can vary, find the value of \(r\), correct to 1 decimal place, for which \(A\) has a stationary value and verify that the flask is most efficient when \(r\) takes this value.

8 The function f is such that \(\mathrm { f } ( x ) = a ^ { 2 } x ^ { 2 } - a x + 3 b\) for \(x \leqslant \frac { 1 } { 2 a }\), where \(a\) and \(b\) are constants.

1. For the case where \(\mathrm { f } ( - 2 ) = 4 a ^ { 2 } - b + 8\) and \(\mathrm { f } ( - 3 ) = 7 a ^ { 2 } - b + 14\), find the possible values of \(a\) and \(b\).
2. For the case where \(a = 1\) and \(b = - 1\), find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and give the domain of \(\mathrm { f } ^ { - 1 }\).

10 \includegraphics[max width=\textwidth, alt={}, center]{f759ce41-708e-4fe7-80b9-adc2be2972ac-18_611_531_262_808} The diagram shows the curve \(y = \mathrm { f } ( x )\) defined for \(x > 0\). The curve has a minimum point at \(A\) and crosses the \(x\)-axis at \(B\) and \(C\). It is given that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 x - \frac { 2 } { x ^ { 3 } }\) and that the curve passes through the point \(\left( 4 , \frac { 189 } { 16 } \right)\).

1. Find the \(x\)-coordinate of \(A\).
2. Find \(\mathrm { f } ( x )\).
3. Find the \(x\)-coordinates of \(B\) and \(C\).
4. Find, showing all necessary working, the area of the shaded region.
   {www.cie.org.uk} after the live examination series. }

8 A curve has equation \(y = x ^ { 3 } + 3 x ^ { 2 } - 9 x + k\), where \(k\) is a constant.

1. Write down an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Find the \(x\)-coordinates of the two stationary points on the curve.
3. Hence find the two values of \(k\) for which the curve has a stationary point on the \(x\)-axis.

8 A solid rectangular block has a base which measures \(2 x \mathrm {~cm}\) by \(x \mathrm {~cm}\). The height of the block is \(y \mathrm {~cm}\) and the volume of the block is \(72 \mathrm {~cm} ^ { 3 }\).

1. Express \(y\) in terms of \(x\) and show that the total surface area, \(A \mathrm {~cm} ^ { 2 }\), of the block is given by $$A = 4 x ^ { 2 } + \frac { 216 } { x }$$ Given that \(x\) can vary,
2. find the value of \(x\) for which \(A\) has a stationary value,
3. find this stationary value and determine whether it is a maximum or a minimum.

10 A curve has equation \(y = x ^ { 2 } + \frac { 2 } { x }\).

1. Write down expressions for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
2. Find the coordinates of the stationary point on the curve and determine its nature.
3. Find the volume of the solid formed when the region enclosed by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 2\) is rotated completely about the \(x\)-axis.

9 \includegraphics[max width=\textwidth, alt={}, center]{dd2cb0ec-5df9-4d99-9e15-5ae1f1c07b96-4_387_903_799_623} The diagram shows an open container constructed out of \(200 \mathrm {~cm} ^ { 2 }\) of cardboard. The two vertical end pieces are isosceles triangles with sides \(5 x \mathrm {~cm} , 5 x \mathrm {~cm}\) and \(8 x \mathrm {~cm}\), and the two side pieces are rectangles of length \(y \mathrm {~cm}\) and width \(5 x \mathrm {~cm}\), as shown. The open top is a horizontal rectangle.

1. Show that \(y = \frac { 200 - 24 x ^ { 2 } } { 10 x }\).
2. Show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the container is given by \(V = 240 x - 28.8 x ^ { 3 }\). Given that \(x\) can vary,
3. find the value of \(x\) for which \(V\) has a stationary value,
4. determine whether it is a maximum or a minimum stationary value.

8 The equation of a curve is \(y = ( 2 x - 3 ) ^ { 3 } - 6 x\).

1. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) in terms of \(x\).
2. Find the \(x\)-coordinates of the two stationary points and determine the nature of each stationary point.

7 \includegraphics[max width=\textwidth, alt={}, center]{08729aab-586b-4210-94c9-77b1f6b1d873-3_385_360_1379_561} \includegraphics[max width=\textwidth, alt={}, center]{08729aab-586b-4210-94c9-77b1f6b1d873-3_364_369_1379_1219} A wire, 80 cm long, is cut into two pieces. One piece is bent to form a square of side \(x \mathrm {~cm}\) and the other piece is bent to form a circle of radius \(r \mathrm {~cm}\) (see diagram). The total area of the square and the circle is \(A \mathrm {~cm} ^ { 2 }\).

1. Show that \(A = \frac { ( \pi + 4 ) x ^ { 2 } - 160 x + 1600 } { \pi }\).
2. Given that \(x\) and \(r\) can vary, find the value of \(x\) for which \(A\) has a stationary value.

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

1. Find the coordinates of the stationary point on the curve and determine its nature.
2. Find the area of the region enclosed by the curve, the \(x\)-axis and the lines \(x = 0\) and \(x = 1\).

6 \includegraphics[max width=\textwidth, alt={}, center]{b566719c-216e-41e5-8431-da77e1dad73e-2_590_666_1720_737} In the diagram, \(O A B C D E F G\) is a cube in which each side has length 6 . Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(\overrightarrow { O A } , \overrightarrow { O C }\) and \(\overrightarrow { O D }\) respectively. The point \(P\) is such that \(\overrightarrow { A P } = \frac { 1 } { 3 } \overrightarrow { A B }\) and the point \(Q\) is the mid-point of \(D F\).

1. Express each of the vectors \(\overrightarrow { O Q }\) and \(\overrightarrow { P Q }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
2. Find the angle \(O Q P\).

8 \includegraphics[max width=\textwidth, alt={}, center]{73c0c113-8f35-4e7f-ad5d-604602498b71-3_314_803_751_671} The diagram shows a metal plate consisting of a rectangle with sides \(x \mathrm {~cm}\) and \(y \mathrm {~cm}\) and a quarter-circle of radius \(x \mathrm {~cm}\). The perimeter of the plate is 60 cm .

1. Express \(y\) in terms of \(x\).
2. Show that the area of the plate, \(A \mathrm {~cm} ^ { 2 }\), is given by \(A = 30 x - x ^ { 2 }\). Given that \(x\) can vary,
3. find the value of \(x\) at which \(A\) is stationary,
4. find this stationary value of \(A\), and determine whether it is a maximum or a minimum value.

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

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and determine, with a reason, whether the curve has any stationary points.
2. Find the volume obtained when the region bounded by the curve, the coordinate axes and the line \(x = 1\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
3. Find the set of values of \(k\) for which the line \(y = x + k\) intersects the curve at two distinct points.

10 \includegraphics[max width=\textwidth, alt={}, center]{ae57d8f1-5a0d-426c-952d-e8b99c6aeaba-4_433_969_1475_587} The diagram shows an open rectangular tank of height \(h\) metres covered with a lid. The base of the tank has sides of length \(x\) metres and \(\frac { 1 } { 2 } x\) metres and the lid is a rectangle with sides of length \(\frac { 5 } { 4 } x\) metres and \(\frac { 4 } { 5 } x\) metres. When full the tank holds \(4 \mathrm {~m} ^ { 3 }\) of water. The material from which the tank is made is of negligible thickness. The external surface area of the tank together with the area of the top of the lid is \(A \mathrm {~m} ^ { 2 }\).

1. Express \(h\) in terms of \(x\) and hence show that \(A = \frac { 3 } { 2 } x ^ { 2 } + \frac { 24 } { x }\).
2. Given that \(x\) can vary, find the value of \(x\) for which \(A\) is a minimum, showing clearly that \(A\) is a minimum and not a maximum.
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5 A curve has equation \(y = \frac { 1 } { x - 3 } + x\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
2. Find the coordinates of the maximum point \(A\) and the minimum point \(B\) on the curve.

4 A function f is defined for \(x \in \mathbb { R }\) and is such that \(\mathrm { f } ^ { \prime } ( x ) = 2 x - 6\). The range of the function is given by \(\mathrm { f } ( x ) \geqslant - 4\).

1. State the value of \(x\) for which \(\mathrm { f } ( x )\) has a stationary value.
2. Find an expression for \(\mathrm { f } ( x )\) in terms of \(x\).

7 \includegraphics[max width=\textwidth, alt={}, center]{56d376c5-b91f-488d-89e2-18edcb14052d-3_534_895_255_625} The diagram shows the dimensions in metres of an L-shaped garden. The perimeter of the garden is 48 m .

1. Find an expression for \(y\) in terms of \(x\).
2. Given that the area of the garden is \(A \mathrm {~m} ^ { 2 }\), show that \(A = 48 x - 8 x ^ { 2 }\).
3. Given that \(x\) can vary, find the maximum area of the garden, showing that this is a maximum value rather than a minimum value.

8 The equation of a curve is \(y = \sqrt { } \left( 8 x - x ^ { 2 } \right)\). Find

1. an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), and the coordinates of the stationary point on the curve,
2. the volume obtained when the region bounded by the curve and the \(x\)-axis is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.

5 A curve has equation \(y = 2 x + \frac { 1 } { ( x - 1 ) ^ { 2 } }\). Verify that the curve has a stationary point at \(x = 2\) and determine its nature.

10 A curve is defined for \(x > 0\) and is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = x + \frac { 4 } { x ^ { 2 } }\). The point \(P ( 4,8 )\) lies on the curve.

1. Find the equation of the curve.
2. Show that the gradient of the curve has a minimum value when \(x = 2\) and state this minimum value.