1.07n Stationary points: find maxima, minima using derivatives

5 \includegraphics[max width=\textwidth, alt={}, center]{0a45a806-007f-4840-85e7-16d4c1a2c599-3_528_757_251_694} The diagram shows the curve \(y = 4 e ^ { \frac { 1 } { 2 } x } - 6 x + 3\) and its minimum point \(M\).

1. Show that the \(x\)-coordinate of \(M\) can be written in the form \(\ln a\), where the value of \(a\) is to be stated.
2. Find the exact value of the area of the region enclosed by the curve and the lines \(x = 0 , x = 2\) and \(y = 0\).

7

1. Find the exact area of the region bounded by the curve \(y = 1 + \mathrm { e } ^ { 2 x - 1 }\), the \(x\)-axis and the lines \(x = \frac { 1 } { 2 }\) and \(x = 2\).
2. \includegraphics[max width=\textwidth, alt={}, center]{e3ee4932-8219-4332-9cd2-e7f835522469-3_469_719_397_753} The diagram shows the curve \(y = \frac { \mathrm { e } ^ { 2 x } } { \sin 2 x }\) for \(0 < x < \frac { 1 } { 2 } \pi\), and its minimum point \(M\). Find the exact \(x\)-coordinate of \(M\).

7 The equation of a curve is $$2 x ^ { 2 } + 3 x y + y ^ { 2 } = 3$$

1. Find the equation of the tangent to the curve at the point \(( 2 , - 1 )\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
2. Show that the curve has no stationary points.

8 \includegraphics[max width=\textwidth, alt={}, center]{de8af872-9f77-4787-8e66-ed199405ca25-3_581_650_1272_744} The diagram shows the curve $$y = \tan x \cos 2 x , \text { for } 0 \leqslant x < \frac { 1 } { 2 } \pi$$ and its maximum point \(M\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 4 \cos ^ { 2 } x - \sec ^ { 2 } x - 2\).
2. Hence find the \(x\)-coordinate of \(M\), giving your answer correct to 2 decimal places.

8 \includegraphics[max width=\textwidth, alt={}, center]{22ba6cc7-7375-434e-9eaa-d536684dd727-3_581_650_1272_744} The diagram shows the curve $$y = \tan x \cos 2 x , \text { for } 0 \leqslant x < \frac { 1 } { 2 } \pi$$ and its maximum point \(M\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 4 \cos ^ { 2 } x - \sec ^ { 2 } x - 2\).
2. Hence find the \(x\)-coordinate of \(M\), giving your answer correct to 2 decimal places.

4 The polynomials \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) are defined by $$\mathrm { f } ( x ) = x ^ { 3 } + a x ^ { 2 } + b \quad \text { and } \quad \mathrm { g } ( x ) = x ^ { 3 } + b x ^ { 2 } - a$$ where \(a\) and \(b\) are constants. It is given that ( \(x + 2\) ) is a factor of \(\mathrm { f } ( x )\). It is also given that, when \(\mathrm { g } ( x )\) is divided by \(( x + 1 )\), the remainder is - 18 .

1. Find the values of \(a\) and \(b\).
2. When \(a\) and \(b\) have these values, find the greatest possible value of \(\mathrm { g } ( x ) - \mathrm { f } ( x )\) as \(x\) varies.

4 \includegraphics[max width=\textwidth, alt={}, center]{3b217eb4-3bd3-4800-a913-749754bf109f-2_524_625_1425_758} The diagram shows the curve \(y = \mathrm { e } ^ { x } + 4 \mathrm { e } ^ { - 2 x }\) and its minimum point \(M\).

1. Show that the \(x\)-coordinate of \(M\) is \(\ln 2\).
2. The region shaded in the diagram is enclosed by the curve and the lines \(x = 0 , x = \ln 2\) and \(y = 0\). Use integration to show that the area of the shaded region is \(\frac { 5 } { 2 }\).

8 \includegraphics[max width=\textwidth, alt={}, center]{6295873e-7db4-4e7e-8dcd-912ad9c41675-10_643_414_260_863} The diagram shows the curve with equation $$y = 3 x ^ { 2 } \ln \left( \frac { 1 } { 6 } x \right) .$$ The curve crosses the \(x\)-axis at the point \(P\) and has a minimum point \(M\).

1. Find the gradient of the curve at the point \(P\).
2. Find the exact coordinates of the point \(M\).

8 \includegraphics[max width=\textwidth, alt={}, center]{de2f8bf3-fd03-4199-9eb2-c9cbac4d4385-10_549_495_258_824} The diagram shows the curve with parametric equations $$x = 2 - \cos 2 t , \quad y = 2 \sin ^ { 3 } t + 3 \cos ^ { 3 } t + 1$$ for \(0 \leqslant t \leqslant \frac { 1 } { 2 } \pi\). The end-points of the curve \(( 1,4 )\) and \(( 3,3 )\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 } { 2 } \sin t - \frac { 9 } { 4 } \cos t\).
2. Find the coordinates of the minimum point, giving each coordinate correct to 3 significant figures.
3. Find the exact gradient of the normal to the curve at the point for which \(x = 2\).

4 \includegraphics[max width=\textwidth, alt={}, center]{873a104f-e2e2-49bb-b943-583769728fbb-06_355_839_260_653} The diagram shows the curve with equation \(y = \frac { 5 \ln x } { 2 x + 1 }\). The curve crosses the \(x\)-axis at the point \(P\) and has a maximum point \(M\).

1. Find the gradient of the curve at the point \(P\).
2. Show that the \(x\)-coordinate of the point \(M\) satisfies the equation \(x = \frac { x + 0.5 } { \ln x }\).
3. Use an iterative formula based on the equation in part (ii) to find the \(x\)-coordinate of \(M\) correct to 4 significant figures. Show the result of each iteration to 6 significant figures.

2 A curve has equation \(y = 3 \ln ( 2 x + 9 ) - 2 \ln x\).

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

5 The equation of a curve is \(y = 2 \cos x + \sin 2 x\). Find the \(x\)-coordinates of the stationary points on the curve for which \(0 < x < \pi\), and determine the nature of each of these stationary points.

8 The equation of a curve is \(y = \ln x + \frac { 2 } { x }\), where \(x > 0\).

1. Find the coordinates of the stationary point of the curve and determine whether it is a maximum or a minimum point.
2. The sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 2 } { 3 - \ln x _ { n } }$$ with initial value \(x _ { 1 } = 1\), converges to \(\alpha\). State an equation satisfied by \(\alpha\), and hence show that \(\alpha\) is the \(x\)-coordinate of a point on the curve where \(y = 3\).
3. Use this iterative formula to find \(\alpha\) correct to 2 decimal places, showing the result of each iteration.

10 \includegraphics[max width=\textwidth, alt={}, center]{2718ebbb-29e3-46f7-8d8d-ec7d526483f8-3_458_920_1144_609} The diagram shows the curve \(y = \frac { \ln x } { x ^ { 2 } }\) and its maximum point \(M\). The curve cuts the \(x\)-axis at \(A\).

1. Write down the \(x\)-coordinate of \(A\).
2. Find the exact coordinates of \(M\).
3. Use integration by parts to find the exact area of the shaded region enclosed by the curve, the \(x\)-axis and the line \(x = \mathrm { e }\).

9 \includegraphics[max width=\textwidth, alt={}, center]{208eab3e-a78c-43b4-918f-a9efc9b4f47a-4_429_748_264_699} The diagram shows part of the curve \(y = \frac { x } { x ^ { 2 } + 1 }\) and its maximum point \(M\). The shaded region \(R\) is bounded by the curve and by the lines \(y = 0\) and \(x = p\).

1. Calculate the \(x\)-coordinate of \(M\).
2. Find the area of \(R\) in terms of \(p\).
3. Hence calculate the value of \(p\) for which the area of \(R\) is 1 , giving your answer correct to 3 significant figures.

9 \includegraphics[max width=\textwidth, alt={}, center]{20893bfc-3300-4205-9d2c-729cc3243971-4_547_1401_264_370} The diagram shows the curve \(y = \mathrm { e } ^ { - \frac { 1 } { 2 } x } \sqrt { } ( 1 + 2 x )\) and its maximum point \(M\). The shaded region between the curve and the axes is denoted by \(R\).

1. Find the \(x\)-coordinate of \(M\).
2. Find by integration the volume of the solid obtained when \(R\) is rotated completely about the \(x\)-axis. Give your answer in terms of \(\pi\) and e.

10 \includegraphics[max width=\textwidth, alt={}, center]{0f73e750-18a0-49ad-b4cb-fd6d14f0789e-4_424_713_262_715} The diagram shows the curve \(y = x ^ { 2 } \sqrt { } \left( 1 - x ^ { 2 } \right)\) for \(x \geqslant 0\) and its maximum point \(M\).

1. Find the exact value of the \(x\)-coordinate of \(M\).
2. Show, by means of the substitution \(x = \sin \theta\), that the area \(A\) of the shaded region between the curve and the \(x\)-axis is given by $$A = \frac { 1 } { 4 } \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sin ^ { 2 } 2 \theta \mathrm {~d} \theta$$
3. Hence obtain the exact value of \(A\).

4 \includegraphics[max width=\textwidth, alt={}, center]{20de5ba6-9426-4431-99af-6e8e62607f3e-2_513_895_1055_625} The diagram shows the curve \(y = \frac { \sin x } { x }\) for \(0 < x \leqslant 2 \pi\), and its minimum point \(M\).

1. Show that the \(x\)-coordinate of \(M\) satisfies the equation $$x = \tan x$$
2. The iterative formula $$x _ { n + 1 } = \tan ^ { - 1 } \left( x _ { n } \right) + \pi$$ can be used to determine the \(x\)-coordinate of \(M\). Use this formula to determine the \(x\)-coordinate of \(M\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

2 The curve \(y = \frac { \ln x } { x ^ { 3 } }\) has one stationary point. Find the \(x\)-coordinate of this point.

8 \includegraphics[max width=\textwidth, alt={}, center]{5b219e1c-e5a0-4f75-910d-fca9761e5088-3_435_895_799_625} The diagram shows the curve \(y = 5 \sin ^ { 3 } x \cos ^ { 2 } x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\), and its maximum point \(M\).

1. Find the \(x\)-coordinate of \(M\).
2. Using the substitution \(u = \cos x\), find by integration the area of the shaded region bounded by the curve and the \(x\)-axis.

5 \includegraphics[max width=\textwidth, alt={}, center]{4c71f68a-efb9-4408-bf03-874e0d4426d5-2_458_807_1786_667} The diagram shows the curve $$y = 8 \sin \frac { 1 } { 2 } x - \tan \frac { 1 } { 2 } x$$ for \(0 \leqslant x < \pi\). The \(x\)-coordinate of the maximum point is \(\alpha\) and the shaded region is enclosed by the curve and the lines \(x = \alpha\) and \(y = 0\).

1. Show that \(\alpha = \frac { 2 } { 3 } \pi\).
2. Find the exact value of the area of the shaded region.

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 with equation \(y = \frac { \mathrm { e } ^ { 2 x } } { x ^ { 3 } }\) has one stationary point.

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

9 \includegraphics[max width=\textwidth, alt={}, center]{436d891d-92ee-4076-8369-db756d413979-3_307_601_1553_772} The diagram shows the curve \(y = \sin ^ { 2 } 2 x \cos x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\), and its maximum point \(M\).

1. Find the \(x\)-coordinate of \(M\).
2. Using the substitution \(u = \sin x\), find by integration the area of the shaded region bounded by the curve and the \(x\)-axis.

10 \includegraphics[max width=\textwidth, alt={}, center]{b6bede75-3da4-4dda-9303-a5a692fc2572-3_556_1093_1596_523} The diagram shows the curve \(y = 10 e ^ { - \frac { 1 } { 2 } x } \sin 4 x\) for \(x \geqslant 0\). The stationary points are labelled \(T _ { 1 } , T _ { 2 }\), \(T _ { 3 } , \ldots\) as shown.

1. Find the \(x\)-coordinates of \(T _ { 1 }\) and \(T _ { 2 }\), giving each \(x\)-coordinate correct to 3 decimal places.
2. It is given that the \(x\)-coordinate of \(T _ { n }\) is greater than 25 . Find the least possible value of \(n\).