1.07n Stationary points: find maxima, minima using derivatives

12 \includegraphics[max width=\textwidth, alt={}, center]{9803d51b-215e-4d03-884f-a67fb7ed6442-20_524_972_274_548} 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. Calculate 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 \(m\). Calculate the value of \(m\).

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.

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

1. Express \(8 x - x ^ { 2 }\) in the form \(a - ( x + b ) ^ { 2 }\), stating the numerical values of \(a\) and \(b\).
2. Hence, or otherwise, find the coordinates of the stationary point of the curve.
3. Find the set of values of \(x\) for which \(y \geqslant - 20\). The function g is defined by \(\mathrm { g } : x \mapsto 8 x - x ^ { 2 }\), for \(x \geqslant 4\).
4. State the domain and range of \(\mathrm { g } ^ { - 1 }\).
5. Find an expression, in terms of \(x\), for \(\mathrm { g } ^ { - 1 } ( x )\).

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 \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.

10 The equation of a curve is \(y = 2 x + \frac { 8 } { x ^ { 2 } }\).

1. Obtain 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 the nature of the stationary point.
3. Show that the normal to the curve at the point \(( - 2 , - 2 )\) intersects the \(x\)-axis at the point \(( - 10,0 )\).
4. Find the area of the region enclosed by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 2\).

4 The equation of a curve \(C\) is \(y = 2 x ^ { 2 } - 8 x + 9\) and the equation of a line \(L\) is \(x + y = 3\).

1. Find the \(x\)-coordinates of the points of intersection of \(L\) and \(C\).
2. Show that one of these points is also the stationary point of \(C\).

11 \includegraphics[max width=\textwidth, alt={}, center]{3b527397-7781-41e9-8218-57277cc977bf-4_686_805_950_669} The diagram shows the curve \(y = x ^ { 3 } - 6 x ^ { 2 } + 9 x\) for \(x \geqslant 0\). The curve has a maximum point at \(A\) and a minimum point on the \(x\)-axis at \(B\). The normal to the curve at \(C ( 2,2 )\) meets the normal to the curve at \(B\) at the point \(D\).

1. Find the coordinates of \(A\) and \(B\).
2. Find the equation of the normal to the curve at \(C\).
3. Find the area of the shaded region.

6 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { \frac { 1 } { 2 } } - 6\) and the point \(( 9,2 )\) lies on the curve.

1. Find the equation of the curve.
2. Find the \(x\)-coordinate of the stationary point on the curve and determine the nature of the stationary point.

9 \includegraphics[max width=\textwidth, alt={}, center]{71fe6352-e0dc-4c3a-8b54-99709a1782ca-4_602_899_248_625} The diagram shows part of the curve \(y = x + \frac { 4 } { x }\) which has a minimum point at \(M\). The line \(y = 5\) intersects the curve at the points \(A\) and \(B\).

1. Find the coordinates of \(A , B\) and \(M\).
2. Find the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.

6 The variables \(x , y\) and \(z\) can take only positive values and are such that $$z = 3 x + 2 y \quad \text { and } \quad x y = 600 .$$

1. Show that \(z = 3 x + \frac { 1200 } { x }\).
2. Find the stationary value of \(z\) and determine its nature.

9 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 } { \sqrt { } x } - 1\) and \(P ( 9,5 )\) is a point on the curve.

1. Find the equation of the curve.
2. Find the coordinates of the stationary point on the curve.
3. Find an expression for \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and determine the nature of the stationary point.
4. The normal to the curve at \(P\) makes an angle of \(\tan ^ { - 1 } k\) with the positive \(x\)-axis. Find the value of \(k\).

10 It is given that a curve has equation \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = x ^ { 3 } - 2 x ^ { 2 } + x\).

1. Find the set of values of \(x\) for which the gradient of the curve is less than 5 .
2. Find the values of \(\mathrm { f } ( x )\) at the two stationary points on the curve and determine the nature of each stationary point.

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.

6 The non-zero variables \(x , y\) and \(u\) are such that \(u = x ^ { 2 } y\). Given that \(y + 3 x = 9\), find the stationary value of \(u\) and determine whether this is a maximum or a minimum value.

12 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { \frac { 1 } { 2 } } - x ^ { - \frac { 1 } { 2 } }\). The curve passes through the point \(\left( 4 , \frac { 2 } { 3 } \right)\).

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 and determine its nature.

8 The equation of a curve is such that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 2 x - 1\). Given that the curve has a minimum point at \(( 3 , - 10 )\), find the coordinates of the maximum point.

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 \(y = \mathrm { g } ( x )\).

5 \includegraphics[max width=\textwidth, alt={}, center]{b6ae63ce-a8a8-45ef-9c75-2fab30de8ad9-2_364_625_1873_762} A farmer divides a rectangular piece of land into 8 equal-sized rectangular sheep pens as shown in the diagram. Each sheep pen measures \(x \mathrm {~m}\) by \(y \mathrm {~m}\) and is fully enclosed by metal fencing. The farmer uses 480 m of fencing.

1. Show that the total area of land used for the sheep pens, \(A \mathrm {~m} ^ { 2 }\), is given by $$A = 384 x - 9.6 x ^ { 2 }$$
2. Given that \(x\) and \(y\) can vary, find the dimensions of each sheep pen for which the value of \(A\) is a maximum. (There is no need to verify that the value of \(A\) is a maximum.)

10 \includegraphics[max width=\textwidth, alt={}, center]{616a6177-0d5c-49f7-b0c1-9138a13c1963-4_687_488_262_826} The diagram shows the part of the curve \(y = \frac { 8 } { x } + 2 x\) for \(x > 0\), and the minimum point \(M\).

1. Find expressions for \(\frac { \mathrm { d } y } { \mathrm {~d} x } , \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and \(\int y ^ { 2 } \mathrm {~d} x\).
2. Find the coordinates of \(M\) and determine the coordinates and nature of the stationary point on the part of the curve for which \(x < 0\).
3. Find the volume obtained when the region bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 2\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.

5 A curve has equation \(y = 8 x + ( 2 x - 1 ) ^ { - 1 }\). Find the values of \(x\) at which the curve has a stationary point and determine the nature of each stationary point, justifying your answers.

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.