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

2 The function f is defined by \(\mathrm { f } ( x ) = \frac { 1 } { 3 } ( 2 x - 1 ) ^ { \frac { 3 } { 2 } } - 2 x\) for \(\frac { 1 } { 2 } < x < a\). It is given that f is a decreasing function. Find the maximum possible value of the constant \(a\).

10 The equation of a curve is such that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 6 x ^ { 2 } - \frac { 4 } { x ^ { 3 } }\). The curve has a stationary point at \(\left( - 1 , \frac { 9 } { 2 } \right)\).

1. Determine the nature of the stationary point at \(\left( - 1 , \frac { 9 } { 2 } \right)\).
2. Find the equation of the curve.
3. Show that the curve has no other stationary points.
4. A point \(A\) is moving along the curve and the \(y\)-coordinate of \(A\) is increasing at a rate of 5 units per second. Find the rate of increase of the \(x\)-coordinate of \(A\) at the point where \(x = 1\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

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

1. Find the coordinates of the stationary point.
2. Determine the nature of the stationary point.
3. For positive values of \(x\), determine whether the curve shows a function that is increasing, decreasing or neither. Give a reason for your answer.

1 The function f is defined by \(\mathrm { f } ( x ) = \frac { 1 } { 3 x + 2 } + x ^ { 2 }\) for \(x < - 1\).
Determine whether f is an increasing function, a decreasing function or neither.

11 It is given that a curve has equation \(y = k ( 3 x - k ) ^ { - 1 } + 3 x\), where \(k\) is a constant.

1. Find, in terms of \(k\), the values of \(x\) at which there is a stationary point.
   The function f has a stationary value at \(x = a\) and is defined by $$f ( x ) = 4 ( 3 x - 4 ) ^ { - 1 } + 3 x \quad \text { for } x \geqslant \frac { 3 } { 2 }$$
2. Find the value of \(a\) and determine the nature of the stationary value.
3. The function g is defined by \(\mathrm { g } ( x ) = - ( 3 x + 1 ) ^ { - 1 } + 3 x\) for \(x \geqslant 0\). Determine, making your reasoning clear, whether \(g\) is an increasing function, a decreasing function or neither.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

3

1. Express \(5 y ^ { 2 } - 30 y + 50\) in the form \(5 ( y + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants.
2. The function f is defined by \(\mathrm { f } ( x ) = x ^ { 5 } - 10 x ^ { 3 } + 50 x\) for \(x \in \mathbb { R }\). Determine whether \(f\) is an increasing function, a decreasing function or neither.

8 The equation of a curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { \frac { 1 } { 2 } } - 3 x ^ { - \frac { 1 } { 2 } }\). The curve passes through the point \(( 3,5 )\).

1. Find the equation of the curve.
2. Find the \(x\)-coordinate of the stationary point.
3. State the set of values of \(x\) for which \(y\) increases as \(x\) increases.

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.

10 The equation of a curve is \(y = x ^ { 2 } - 3 x + 4\).

1. Show that the whole of the curve lies above the \(x\)-axis.
2. Find the set of values of \(x\) for which \(x ^ { 2 } - 3 x + 4\) is a decreasing function of \(x\). The equation of a line is \(y + 2 x = k\), where \(k\) is a constant.
3. In the case where \(k = 6\), find the coordinates of the points of intersection of the line and the curve.
4. Find the value of \(k\) for which the line is a tangent to the curve.

10 \includegraphics[max width=\textwidth, alt={}, center]{cbcb15b4-1870-4dfd-b6e9-839aa4601511-4_515_885_662_630} The diagram shows the curve \(y = x ^ { 3 } - 3 x ^ { 2 } - 9 x + k\), where \(k\) is a constant. The curve has a minimum point on the \(x\)-axis.

1. Find the value of \(k\).
2. Find the coordinates of the maximum point of the curve.
3. State the set of values of \(x\) for which \(x ^ { 3 } - 3 x ^ { 2 } - 9 x + k\) is a decreasing function of \(x\).
4. Find the area of the shaded region.

1 It is given that \(\mathrm { f } ( x ) = ( 2 x - 5 ) ^ { 3 } + x\), for \(x \in \mathbb { R }\). Show that f is an increasing function.

9 A curve has equation \(y = \mathrm { f } ( x )\) and is such that \(\mathrm { f } ^ { \prime } ( x ) = 3 x ^ { \frac { 1 } { 2 } } + 3 x ^ { - \frac { 1 } { 2 } } - 10\).

1. By using the substitution \(u = x ^ { \frac { 1 } { 2 } }\), or otherwise, find the values of \(x\) for which the curve \(y = \mathrm { f } ( x )\) has stationary points.
2. Find \(\mathrm { f } ^ { \prime \prime } ( x )\) and hence, or otherwise, determine the nature of each stationary point.
3. It is given that the curve \(y = \mathrm { f } ( x )\) passes through the point \(( 4 , - 7 )\). Find \(\mathrm { f } ( x )\).

8 The volume of a solid circular cylinder of radius \(r \mathrm {~cm}\) is \(250 \pi \mathrm {~cm} ^ { 3 }\).

1. Show that the total surface area, \(S \mathrm {~cm} ^ { 2 }\), of the cylinder is given by $$S = 2 \pi r ^ { 2 } + \frac { 500 \pi } { r }$$
2. Given that \(r\) can vary, find the stationary value of \(S\).
3. Determine the nature of this stationary value.

9 A function f is defined by \(\mathrm { f } ( x ) = \frac { 5 } { 1 - 3 x }\), for \(x \geqslant 1\).

1. Find an expression for \(\mathrm { f } ^ { \prime } ( x )\).
2. Determine, with a reason, whether \(f\) is an increasing function, a decreasing function or neither.
3. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\), and state the domain and range of \(\mathrm { f } ^ { - 1 }\).

9 The base of a cuboid has sides of length \(x \mathrm {~cm}\) and \(3 x \mathrm {~cm}\). The volume of the cuboid is \(288 \mathrm {~cm} ^ { 3 }\).

1. Show that the total surface area of the cuboid, \(A \mathrm {~cm} ^ { 2 }\), is given by $$A = 6 x ^ { 2 } + \frac { 768 } { x }$$
2. Given that \(x\) can vary, find the stationary value of \(A\) and determine its nature.

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

1. Find \(\mathrm { f } ^ { \prime } ( x )\).
2. State, with a reason, whether f is an increasing function, a decreasing function or neither. The function g is defined by \(\mathrm { g } ( x ) = \frac { 1 } { x + 1 } + \frac { 1 } { ( x + 1 ) ^ { 2 } }\) for \(x < - 1\).
3. Find the coordinates of the stationary point on the curve \(y = \mathrm { g } ( x )\).

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.

9 The equation of a curve is \(y = 8 \sqrt { } x - 2 x\).

1. Find the coordinates of the stationary point of the curve.
2. Find an expression for \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and hence, or otherwise, determine the nature of the stationary point.
3. Find the values of \(x\) at which the line \(y = 6\) meets the curve.
4. State the set of values of \(k\) for which the line \(y = k\) does not meet the curve.

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.

8

1. The tangent to the curve \(y = x ^ { 3 } - 9 x ^ { 2 } + 24 x - 12\) at a point \(A\) is parallel to the line \(y = 2 - 3 x\). Find the equation of the tangent at \(A\).
2. The function f is defined by \(\mathrm { f } ( x ) = x ^ { 3 } - 9 x ^ { 2 } + 24 x - 12\) for \(x > k\), where \(k\) is a constant. Find the smallest value of \(k\) for f to be an increasing function.

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.

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.