1.07p Points of inflection: using second derivative

11 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 ( 3 x - 5 ) ^ { 3 } - k x ^ { 2 }\), where \(k\) is a constant. The curve has a stationary point at \(( 2 , - 3.5 )\).

1. Find the value of \(k\).
   ................................................................................................................................................. . .
2. Find the equation of the curve.
3. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
4. Determine the nature of the stationary point at \(( 2 , - 3.5 )\).

9 The equation of a curve is \(y = 3 x + 1 - 4 ( 3 x + 1 ) ^ { \frac { 1 } { 2 } }\) for \(x > - \frac { 1 } { 3 }\).

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 stationary point of the curve and determine its nature.

11 The equation of a curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x ^ { 2 } - 30 x + 6 a\), where \(a\) is a positive constant. The curve has a stationary point at \(( a , - 15 )\).

1. Find the value of \(a\).
2. Determine the nature of this stationary point.
3. Find the equation of the curve.
4. Find the coordinates of any other stationary points on the curve.

10 The gradient of a curve at the point \(( x , y )\) is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 ( x + 3 ) ^ { \frac { 1 } { 2 } } - x\). The curve has a stationary point at \(( a , 14 )\), where \(a\) is a positive constant.

1. Find the value of \(a\).
2. Determine the nature of the stationary point.
3. Find the equation of the curve.

9 A curve has equation \(y = \mathrm { f } ( x )\), and it is given that \(\mathrm { f } ^ { \prime } ( x ) = 2 x ^ { 2 } - 7 - \frac { 4 } { x ^ { 2 } }\).

1. Given that \(\mathrm { f } ( 1 ) = - \frac { 1 } { 3 }\), find \(\mathrm { f } ( x )\).
2. Find the coordinates of the stationary points on the curve.
3. Find \(\mathrm { f } ^ { \prime \prime } ( x )\).
4. Hence, or otherwise, determine the nature of each of the stationary points.

10 The function f is defined by \(\mathrm { f } ( x ) = x ^ { 2 } + \frac { k } { x } + 2\) for \(x > 0\).

1. Given that the curve with equation \(y = \mathrm { f } ( x )\) has a stationary point when \(x = 2\), find \(k\).
2. Determine the nature of the stationary point.
3. Given that this is the only stationary point of the curve, find the range of f .

8 A hollow circular cylinder, open at one end, is constructed of thin sheet metal. The total external surface area of the cylinder is \(192 \pi \mathrm {~cm} ^ { 2 }\). The cylinder has a radius of \(r \mathrm {~cm}\) and a height of \(h \mathrm {~cm}\).

1. Express \(h\) in terms of \(r\) and show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the cylinder is given by $$V = \frac { 1 } { 2 } \pi \left( 192 r - r ^ { 3 } \right) .$$ Given that \(r\) can vary,
2. find the value of \(r\) for which \(V\) has a stationary value,
3. find this stationary value and determine whether it is a maximum or a minimum.

8 \includegraphics[max width=\textwidth, alt={}, center]{22a31966-4433-4d7d-8a75-bcd536acfa24-4_543_511_264_817} The diagram shows a glass window consisting of a rectangle of height \(h \mathrm {~m}\) and width \(2 r \mathrm {~m}\) and a semicircle of radius \(r \mathrm {~m}\). The perimeter of the window is 8 m .

1. Express \(h\) in terms of \(r\).
2. Show that the area of the window, \(A \mathrm {~m} ^ { 2 }\), is given by $$A = 8 r - 2 r ^ { 2 } - \frac { 1 } { 2 } \pi r ^ { 2 } .$$ Given that \(r\) can vary,
3. find the value of \(r\) for which \(A\) has a stationary value,
4. determine whether this stationary value is a maximum or a minimum.

6 The horizontal base of a solid prism is an equilateral triangle of side \(x \mathrm {~cm}\). The sides of the prism are vertical. The height of the prism is \(h \mathrm {~cm}\) and the volume of the prism is \(2000 \mathrm {~cm} ^ { 3 }\).

1. Express \(h\) in terms of \(x\) and show that the total surface area of the prism, \(A \mathrm {~cm} ^ { 2 }\), is given by $$A = \frac { \sqrt { } 3 } { 2 } x ^ { 2 } + \frac { 24000 } { \sqrt { } 3 } x ^ { - 1 }$$
2. Given that \(x\) can vary, find the value of \(x\) for which \(A\) has a stationary value.
3. Determine, showing all necessary working, the nature of this stationary value.

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.

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.

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.

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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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.

6 The curve \(y = 4 x ^ { 2 } \ln x\) has one stationary point.

1. Find the coordinates of this stationary point, giving your answers correct to 3 decimal places.
2. Determine whether this point is a maximum or a minimum point.

6 Find the exact coordinates of the point on the curve \(y = x \mathrm { e } ^ { - \frac { 1 } { 2 } x }\) at which \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 0\).

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

The curve \(C\) has equation \(y = \mathrm { f } ( x ) , x > 0\) Given that

• the point \(P ( 2,8 \sqrt { 2 } )\) lies on \(C\)
• \(\mathrm { f } ^ { \prime } ( x ) = 4 \sqrt { x ^ { 3 } } + \frac { k } { x ^ { 2 } }\) where \(k\) is a constant
• \(\mathrm { f } ^ { \prime \prime } ( x ) = 0\) at \(P\)
  1. find the exact value of \(k\),
  2. find \(\mathrm { f } ( x )\), giving your answer in simplest form.

10. The curve \(C\) has equation $$y = 12 x ^ { \frac { 5 } { 4 } } - \frac { 5 } { 18 } x ^ { 2 } - 1000 , \quad x > 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
2. Hence find the coordinates of the stationary point on \(C\).
3. Use \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) to determine the nature of this stationary point.

15. \begin{figure}[h] \end{figure} Figure 4 shows a solid wooden block. The block is a right prism with length \(h \mathrm {~cm}\). The cross-section of the block is a semi-circle with radius \(r \mathrm {~cm}\). The total surface area of the block, including the curved surface, the two semi-circular ends and the rectangular base, is \(200 \mathrm {~cm} ^ { 2 }\)

1. Show that the volume \(V \mathrm {~cm} ^ { 3 }\) of the block is given by $$V = \frac { \pi r \left( 200 - \pi r ^ { 2 } \right) } { 4 + 2 \pi }$$
2. Use calculus to find the maximum value of \(V\). Give your answer to the nearest \(\mathrm { cm } ^ { 3 }\).
3. Justify, by further differentiation, that the value of \(V\) that you have found is a maximum.

15. \begin{figure}[h] \end{figure} Figure 2 shows a plan for a garden.
The garden consists of two identical rectangles of width \(y \mathrm {~m}\) and length \(x \mathrm {~m}\), joined to a sector of a circle with radius \(x \mathrm {~m}\) and angle 0.8 radians, as shown in Figure 2. The area of the garden is \(60 \mathrm {~m} ^ { 2 }\).

1. Show that the perimeter, \(P \mathrm {~m}\), of the garden is given by $$P = 2 x + \frac { 120 } { x }$$
2. Use calculus to find the exact minimum value for \(P\), giving your answer in the form \(a \sqrt { b }\), where \(a\) and \(b\) are integers.
3. Justify that the value of \(P\) found in part (b) is the minimum. \includegraphics[max width=\textwidth, alt={}, center]{1f61f78b-5e77-4758-8ad5-ea00c7dfea2b-49_83_59_2636_1886}

10. A curve \(C\) has equation $$y = 4 x ^ { 3 } - 9 x + \frac { k } { x } \quad x > 0$$ where \(k\) is a constant.
The point \(P\) with \(x\) coordinate \(\frac { 1 } { 2 }\) lies on \(C\).
Given that \(P\) is a stationary point of \(C\),

1. show that \(k = - \frac { 3 } { 2 }\)
2. Determine the nature of the stationary point at \(P\), justifying your answer. The curve \(C\) has a second stationary point.
3. Using algebra, find the \(x\) coordinate of this second stationary point. \includegraphics[max width=\textwidth, alt={}, center]{08aac50c-7317-4510-927a-7f5f2e00f485-26_2255_50_312_1980}

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. In this question you must show all stages of your working. \section*{Solutions relying entirely on calculator technology are not acceptable.} The curve \(C\) has equation $$y = 27 x ^ { \frac { 1 } { 2 } } - x ^ { \frac { 3 } { 2 } } - 20 \quad x > 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving each term in simplest form.
2. Hence find the coordinates of the stationary point of \(C\).
3. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and hence determine the nature of the stationary point of \(C\).

5. A company makes a particular type of watch. The annual profit made by the company from sales of these watches is modelled by the equation $$P = 12 x - x ^ { \frac { 3 } { 2 } } - 120$$ where \(P\) is the annual profit measured in thousands of pounds and \(\pounds x\) is the selling price of the watch. According to this model,

1. find, using calculus, the maximum possible annual profit.
2. Justify, also using calculus, that the profit you have found is a maximum.