1.07o Increasing/decreasing: functions using sign of dy/dx

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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6 A curve has equation \(y = \mathrm { f } ( x )\). It is given that \(\mathrm { f } ^ { \prime } ( x ) = 3 x ^ { 2 } + 2 x - 5\).

1. Find the set of values of \(x\) for which f is an increasing function.
2. Given that the curve passes through \(( 1,3 )\), find \(\mathrm { f } ( x )\).

2 A curve has equation \(y = 3 x ^ { 3 } - 6 x ^ { 2 } + 4 x + 2\). Show that the gradient of the curve is never negative.

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.

2 It is given that \(\mathrm { f } ( x ) = \frac { 1 } { x ^ { 3 } } - x ^ { 3 }\), for \(x > 0\). Show that f is a decreasing function.

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.

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.

3

1. Express \(3 x ^ { 2 } - 6 x + 2\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants.
2. The function f , where \(\mathrm { f } ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 7 x - 8\), is defined for \(x \in \mathbb { R }\). Find \(\mathrm { f } ^ { \prime } ( x )\) and state, with a reason, whether f is an increasing function, a decreasing function or neither.

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. }

2 A function f is defined by \(\mathrm { f } : x \mapsto x ^ { 3 } - x ^ { 2 } - 8 x + 5\) for \(x < a\). It is given that f is an increasing function. Find the largest possible value of the constant \(a\).

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.

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.

4 The function f is such that \(\mathrm { f } ( x ) = ( 2 x - 1 ) ^ { \frac { 3 } { 2 } } - 6 x\) for \(\frac { 1 } { 2 } < x < k\), where \(k\) is a constant. Find the largest value of \(k\) for which f is a decreasing function.

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.

2 The function f is defined by \(\mathrm { f } ( x ) = x ^ { 3 } + 2 x ^ { 2 } - 4 x + 7\) for \(x \geqslant - 2\). Determine, showing all necessary working, whether f is an increasing function, a decreasing function or neither.

2 An increasing function, f , is defined for \(x > n\), where \(n\) is an integer. It is given that \(\mathrm { f } ^ { \prime } ( x ) = x ^ { 2 } - 6 x + 8\). Find the least possible value of \(n\).

8 A function f is defined for \(x > \frac { 1 } { 2 }\) and is such that \(\mathrm { f } ^ { \prime } ( x ) = 3 ( 2 x - 1 ) ^ { \frac { 1 } { 2 } } - 6\).

1. Find the set of values of \(x\) for which f is decreasing.
2. It is now given that \(\mathrm { f } ( 1 ) = - 3\). Find \(\mathrm { f } ( x )\).

6 The equation of a curve is \(y = 3 \sin x + 4 \cos ^ { 3 } x\).

1. Find the \(x\)-coordinates of the stationary points of the curve in the interval \(0 < x < \pi\).
2. Determine the nature of the stationary point in this interval for which \(x\) is least.

4 The curve \(y = \mathrm { e } ^ { x } + 4 \mathrm { e } ^ { - 2 x }\) has one stationary point.

1. Find the \(x\)-coordinate of this point.
2. Determine whether the stationary point is a maximum or a minimum point.

3 The curve with equation \(y = 6 \mathrm { e } ^ { x } - \mathrm { e } ^ { 3 x }\) has one stationary point.

1. Find the \(x\)-coordinate of this point.
2. Determine whether this point is a maximum or a minimum point.

4 The curve with equation \(y = \frac { 2 - \sin x } { \cos x }\) has one stationary point in the interval \(- \frac { 1 } { 2 } \pi < x < \frac { 1 } { 2 } \pi\).

1. Find the exact coordinates of this point.
2. Determine whether this point is a maximum or a minimum point.

6 The curve \(C\) has equation \(\mathrm { y } = \frac { \mathrm { x } ^ { 2 } + 2 \mathrm { x } - 15 } { \mathrm { x } - 2 }\).

1. Find the equations of the asymptotes of \(C\).
2. Show that \(C\) has no stationary points.
3. Sketch \(C\), stating the coordinates of the intersections with the axes.
4. Sketch the curve with equation \(\mathrm { y } = \left| \frac { \mathrm { x } ^ { 2 } - 2 \mathrm { x } - 15 } { \mathrm { x } - 2 } \right|\).
5. Find the set of values of \(x\) for which \(\left| \frac { 2 x ^ { 2 } + 4 x - 30 } { x - 2 } \right| < 15\).