1.07n Stationary points: find maxima, minima using derivatives

8 A curve is such that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 ( 3 x + 4 ) ^ { \frac { 3 } { 2 } } - 6 x - 8 .$$

1. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
2. Verify that the curve has a stationary point when \(x = - 1\) and determine its nature.
3. It is now given that the stationary point on the curve has coordinates \(( - 1,5 )\). Find the equation of the curve.

11 \includegraphics[max width=\textwidth, alt={}, center]{d3c76ceb-cff7-4155-9697-5c302a9d63a9-4_526_974_822_587} The diagram shows the curve with equation \(y = x ( x - 2 ) ^ { 2 }\). The minimum point on the curve has coordinates \(( a , 0 )\) and the \(x\)-coordinate of the maximum point is \(b\), where \(a\) and \(b\) are constants.

1. State the value of \(a\).
2. Find the value of \(b\).
3. Find the area of the shaded region.
4. The gradient, \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), of the curve has a minimum value \(m\). Find the value of \(m\).

8 \includegraphics[max width=\textwidth, alt={}, center]{02da6b6a-6db1-4bc3-ad4e-537e4f61dcac-3_365_663_1813_740} The inside lane of a school running track consists of two straight sections each of length \(x\) metres, and two semicircular sections each of radius \(r\) metres, as shown in the diagram. The straight sections are perpendicular to the diameters of the semicircular sections. The perimeter of the inside lane is 400 metres.

1. Show that the area, \(A \mathrm {~m} ^ { 2 }\), of the region enclosed by the inside lane is given by \(A = 400 r - \pi r ^ { 2 }\).
2. Given that \(x\) and \(r\) can vary, show that, when \(A\) has a stationary value, there are no straight sections in the track. Determine whether the stationary value is a maximum or a minimum. [5]

6 \includegraphics[max width=\textwidth, alt={}, center]{d5f66324-e1fc-40e1-98e7-625187e24d3d-3_465_663_1160_740} In the diagram, \(S\) is the point ( 0,12 ) and \(T\) is the point ( 16,0 ). The point \(Q\) lies on \(S T\), between \(S\) and \(T\), and has coordinates \(( x , y )\). The points \(P\) and \(R\) lie on the \(x\)-axis and \(y\)-axis respectively and \(O P Q R\) is a rectangle.

1. Show that the area, \(A\), of the rectangle \(O P Q R\) is given by \(A = 12 x - \frac { 3 } { 4 } x ^ { 2 }\).
2. Given that \(x\) can vary, find the stationary value of \(A\) and determine its nature.

9 A curve has equation \(y = \frac { k ^ { 2 } } { x + 2 } + x\), where \(k\) is a positive constant. Find, in terms of \(k\), the values of \(x\) for which the curve has stationary points and determine the nature of each stationary point.

9 The function f is defined for \(x > 0\) and is such that \(\mathrm { f } ^ { \prime } ( x ) = 2 x - \frac { 2 } { x ^ { 2 } }\). The curve \(y = \mathrm { f } ( x )\) passes through the point \(P ( 2,6 )\).

1. Find the equation of the normal to the curve at \(P\).
2. Find the equation of the curve.
3. Find the \(x\)-coordinate of the stationary point and state with a reason whether this point is a maximum or a minimum.

5 A curve has equation \(y = \frac { 8 } { x } + 2 x\).

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 stationary points and state, with a reason, the nature of each stationary point.

9 The curve \(y = \mathrm { f } ( x )\) has a stationary point at \(( 2,10 )\) and it is given that \(\mathrm { f } ^ { \prime \prime } ( x ) = \frac { 12 } { x ^ { 3 } }\).

1. Find \(\mathrm { f } ( x )\).
2. Find the coordinates of the other stationary point.
3. Find the nature of each of the stationary points.

10 The function f is defined by \(\mathrm { f } ( x ) = 2 x + ( x + 1 ) ^ { - 2 }\) for \(x > - 1\).

1. Find \(\mathrm { f } ^ { \prime } ( x )\) and \(\mathrm { f } ^ { \prime \prime } ( x )\) and hence verify that the function f has a minimum value at \(x = 0\). \includegraphics[max width=\textwidth, alt={}, center]{5c1ab2aa-3609-4245-b87a-98ecedc83a11-4_515_920_959_609} The points \(A \left( - \frac { 1 } { 2 } , 3 \right)\) and \(B \left( 1,2 \frac { 1 } { 4 } \right)\) lie on the curve \(y = 2 x + ( x + 1 ) ^ { - 2 }\), as shown in the diagram.
2. Find the distance \(A B\).
3. Find, showing all necessary working, the area of the shaded region. {www.cie.org.uk} after the live examination series. }

10 A curve has equation \(y = \mathrm { f } ( x )\) and it is given that \(\mathrm { f } ^ { \prime } ( x ) = 3 x ^ { \frac { 1 } { 2 } } - 2 x ^ { - \frac { 1 } { 2 } }\). The point \(A\) is the only point on the curve at which the gradient is - 1 .

1. Find the \(x\)-coordinate of \(A\).
2. Given that the curve also passes through the point \(( 4,10 )\), find the \(y\)-coordinate of \(A\), giving your answer as a fraction.

11 The point \(P ( 3,5 )\) lies on the curve \(y = \frac { 1 } { x - 1 } - \frac { 9 } { x - 5 }\).

1. Find the \(x\)-coordinate of the point where the normal to the curve at \(P\) intersects the \(x\)-axis.
2. Find the \(x\)-coordinate of each of the stationary points on the curve and determine the nature of each stationary point, justifying your answers. {www.cie.org.uk} after the live examination series. }

4 Machines in a factory make cardboard cones of base radius \(r \mathrm {~cm}\) and vertical height \(h \mathrm {~cm}\). The volume, \(V \mathrm {~cm} ^ { 3 }\), of such a cone is given by \(V = \frac { 1 } { 3 } \pi r ^ { 2 } h\). The machines produce cones for which \(h + r = 18\).

1. Show that \(V = 6 \pi r ^ { 2 } - \frac { 1 } { 3 } \pi r ^ { 3 }\).
2. Given that \(r\) can vary, find the non-zero value of \(r\) for which \(V\) has a stationary value and show that the stationary value is a maximum.
3. Find the maximum volume of a cone that can be made by these machines.

7 Points \(A\) and \(B\) lie on the curve \(y = x ^ { 2 } - 4 x + 7\). Point \(A\) has coordinates \(( 4,7 )\) and \(B\) is the stationary point of the curve. The equation of a line \(L\) is \(y = m x - 2\), where \(m\) is a constant.

1. In the case where \(L\) passes through the mid-point of \(A B\), find the value of \(m\).
2. Find the set of values of \(m\) for which \(L\) does not meet the curve.

8 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - x ^ { 2 } + 5 x - 4\).

1. Find the \(x\)-coordinate of each of the stationary points of the curve.
2. Obtain an expression for \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and hence or otherwise find the nature of each of the stationary points.
3. Given that the curve passes through the point \(( 6,2 )\), find the equation of the curve.

10 A curve has equation \(y = \mathrm { f } ( x )\) and it is given that \(\mathrm { f } ^ { \prime } ( x ) = a x ^ { 2 } + b x\), where \(a\) and \(b\) are positive constants.

1. Find, in terms of \(a\) and \(b\), the non-zero value of \(x\) for which the curve has a stationary point and determine, showing all necessary working, the nature of the stationary point.
2. It is now given that the curve has a stationary point at \(( - 2 , - 3 )\) and that the gradient of the curve at \(x = 1\) is 9 . Find \(\mathrm { f } ( x )\).

6 A curve has a stationary point at \(\left( 3,9 \frac { 1 } { 2 } \right)\) and has an equation for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = a x ^ { 2 } + a ^ { 2 } x\), where \(a\) is a non-zero constant.

1. Find the value of \(a\). \includegraphics[max width=\textwidth, alt={}, center]{d178603a-f59a-4986-b5ab-b47eceedb2fc-08_67_1569_461_328}
2. Find the equation of the curve.
3. Determine, showing all necessary working, the nature of the stationary point.

8 A curve passes through \(( 0,11 )\) and has an equation for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = a x ^ { 2 } + b x - 4\), where \(a\) and \(b\) are constants.

1. Find the equation of the curve in terms of \(a\) and \(b\).
2. It is now given that the curve has a stationary point at \(( 2,3 )\). Find the values of \(a\) and \(b\).

9 A curve for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = ( 5 x - 1 ) ^ { \frac { 1 } { 2 } } - 2\) passes through the point ( 2,3 ).

1. Find the equation of the curve.
2. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
3. Find the coordinates of the stationary point on the curve and, showing all necessary working, determine the nature of this stationary point.

5 \includegraphics[max width=\textwidth, alt={}, center]{567c3d72-c633-4ae0-8605-f63f93d718c4-08_512_460_258_772} \includegraphics[max width=\textwidth, alt={}, center]{567c3d72-c633-4ae0-8605-f63f93d718c4-08_462_85_260_1279} The diagram shows a solid cone which has a slant height of 15 cm and a vertical height of \(h \mathrm {~cm}\).

1. Show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the cone is given by \(V = \frac { 1 } { 3 } \pi \left( 225 h - h ^ { 3 } \right)\).
   [0pt] [The volume of a cone of radius \(r\) and vertical height \(h\) is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\).]
2. Given that \(h\) can vary, find the value of \(h\) for which \(V\) has a stationary value. Determine, showing all necessary working, the nature of this stationary value.

3 The equation of a curve is \(y = x ^ { 3 } + x ^ { 2 } - 8 x + 7\). The curve has no stationary points in the interval \(a < x < b\). Find the least possible value of \(a\) and the greatest possible value of \(b\).

5 A curve has equation \(y = \frac { 8 } { x } + 2 x\).

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 stationary points and state, with a reason, the nature of each stationary point.

5 \includegraphics[max width=\textwidth, alt={}, center]{8bdd1285-9e39-465a-8c09-bbe410504f9d-06_442_698_260_721} The diagram shows part of the curve with equation \(y = x ^ { 3 } \cos 2 x\). The curve has a maximum at the point \(M\).

1. Show that the \(x\)-coordinate of \(M\) satisfies the equation \(x = \sqrt [ 3 ] { 1.5 x ^ { 2 } \cot 2 x }\).
2. Use the equation in part (a) to show by calculation that the \(x\)-coordinate of \(M\) lies between 0.59 and 0.60.
3. Use an iterative formula, based on the equation in part (a), to find the \(x\)-coordinate of \(M\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures.

2 Find the exact coordinates of the stationary point on the curve with equation \(y = 5 x \mathrm { e } ^ { \frac { 1 } { 2 } x }\).

6 A curve has equation \(y = \frac { 9 \mathrm { e } ^ { 2 x } + 16 } { \mathrm { e } ^ { x } - 1 }\).

1. Show that the \(x\)-coordinate of any stationary point on the curve satisfies the equation $$\mathrm { e } ^ { x } \left( 3 \mathrm { e } ^ { x } - 8 \right) \left( 3 \mathrm { e } ^ { x } + 2 \right) = 0$$
2. Hence show that the curve has only one stationary point and find its exact coordinates.

5 \includegraphics[max width=\textwidth, alt={}, center]{3966e088-0a2f-434a-94fc-40765cd157a7-06_376_848_269_644} The diagram shows the curve with parametric equations $$x = 4 \mathrm { e } ^ { 2 t } , \quad y = 5 \mathrm { e } ^ { - t } \cos 2 t$$ for \(- \frac { 1 } { 4 } \pi \leqslant t \leqslant \frac { 1 } { 4 } \pi\). The curve has a maximum point \(M\).

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
2. Find the coordinates of \(M\), giving each coordinate correct to 3 significant figures.