1.07n Stationary points: find maxima, minima using derivatives

6 \includegraphics[max width=\textwidth, alt={}, center]{b2136f5d-0d66-4524-bb76-fcc4cb59150c-3_405_914_260_612} The diagram shows the curve \(\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 2 \left( x ^ { 2 } - y ^ { 2 } \right)\) and one of its maximum points \(M\). Find the coordinates of \(M\).

9 \includegraphics[max width=\textwidth, alt={}, center]{b2136f5d-0d66-4524-bb76-fcc4cb59150c-3_639_387_1749_879} The diagram shows the curve \(y = \mathrm { e } ^ { 2 \sin x } \cos x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\), and its maximum point \(M\).

1. Using the substitution \(u = \sin x\), find the exact value of the area of the shaded region bounded by the curve and the axes.
2. Find the \(x\)-coordinate of \(M\), giving your answer correct to 3 decimal places.

4 The equation of a curve is $$y = 3 \cos 2 x + 7 \sin x + 2$$ Find the \(x\)-coordinates of the stationary points in the interval \(0 \leqslant x \leqslant \pi\). Give each answer correct to 3 significant figures.

9 \includegraphics[max width=\textwidth, alt={}, center]{3eefd6c1-924c-4b7e-8d17-a2942fb48234-3_399_696_255_721} The diagram shows the curve \(y = x ^ { 2 } \mathrm { e } ^ { 2 - x }\) and its maximum point \(M\).

1. Show that the \(x\)-coordinate of \(M\) is 2 .
2. Find the exact value of \(\int _ { 0 } ^ { 2 } x ^ { 2 } \mathrm { e } ^ { 2 - x } \mathrm {~d} x\).

3 A curve has equation \(y = \cos x \cos 2 x\). Find the \(x\)-coordinate of the stationary point on the curve in the interval \(0 < x < \frac { 1 } { 2 } \pi\), giving your answer correct to 3 significant figures.

4 The curve with equation \(y = \frac { \mathrm { e } ^ { 2 x } } { 4 + \mathrm { e } ^ { 3 x } }\) has one stationary point. Find the exact values of the coordinates of this point.

5 The curve with equation \(y = \sin x \cos 2 x\) has one stationary point in the interval \(0 < x < \frac { 1 } { 2 } \pi\). Find the \(x\)-coordinate of this point, giving your answer correct to 3 significant figures.

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

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x ^ { 2 } - 2 x y } { x ^ { 2 } - y ^ { 2 } }\).
2. Find the coordinates of the points on the curve where the tangent is parallel to the \(x\)-axis.

4 The curve with equation \(y = \frac { ( \ln x ) ^ { 2 } } { x }\) has two stationary points. Find the exact values of the coordinates of these points.

10 \includegraphics[max width=\textwidth, alt={}, center]{b00cefad-7c3c-4672-b309-f19aafab8b01-18_324_677_259_734} The diagram shows the curve \(y = \sin x \cos ^ { 2 } 2 x\) for \(0 \leqslant x \leqslant \frac { 1 } { 4 } \pi\) and its maximum point \(M\).

1. Using the substitution \(u = \cos x\), find by integration the exact area of the shaded region bounded by the curve and the \(x\)-axis.
2. Find the \(x\)-coordinate of \(M\). Give your answer correct to 2 decimal places.

5 A curve has equation \(y = \frac { 2 } { 3 } \ln \left( 1 + 3 \cos ^ { 2 } x \right)\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\).

1. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\tan x\).
2. Hence find the \(x\)-coordinate of the point on the curve where the gradient is - 1 . Give your answer correct to 3 significant figures.

7 \includegraphics[max width=\textwidth, alt={}, center]{7f6f82c3-37d3-48da-9958-e4ef366a6467-10_389_488_258_831} The diagram shows a sketch of the curve \(y = \frac { \mathrm { e } ^ { \frac { 1 } { 2 } x } } { x }\) for \(x > 0\), and its minimum point \(M\).

1. Find the \(x\)-coordinate of \(M\).
2. Use the trapezium rule with two intervals to estimate the value of $$\int _ { 1 } ^ { 3 } \frac { \mathrm { e } ^ { \frac { 1 } { 2 } x } } { x } \mathrm {~d} x$$ giving your answer correct to 2 decimal places.
3. The estimate found in part (ii) is denoted by \(E\). Explain, without further calculation, whether another estimate found using the trapezium rule with four intervals would be greater than \(E\) or less than \(E\).

10 \includegraphics[max width=\textwidth, alt={}, center]{dcfbe7af-c212-42b1-8a90-8e0418cf0ffd-16_330_689_264_726} The diagram shows the curve \(y = \sin ^ { 3 } x \sqrt { } ( \cos x )\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\), and its maximum point \(M\).

1. Using the substitution \(u = \cos x\), find by integration the exact area of the shaded region bounded by the curve and the \(x\)-axis.
2. Showing all your working, find the \(x\)-coordinate of \(M\), giving your answer correct to 3 decimal places.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

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.

6 \includegraphics[max width=\textwidth, alt={}, center]{79efa364-da5a-4888-85a9-dc4de1e0908e-3_543_825_287_660} The diagram shows the curve \(y = ( 3 - x ) \mathrm { e } ^ { - 2 x }\) and its minimum point \(M\). The curve intersects the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\).

1. Calculate the \(x\)-coordinate of \(M\).
2. Find the area of the region bounded by \(O A , O B\) and the curve, giving your answer in terms of e.

7 \includegraphics[max width=\textwidth, alt={}, center]{8c533469-393c-4e4c-a6ec-eab1303741e7-3_480_901_973_621} The diagram shows the curve \(y = x ^ { 2 } e ^ { - \frac { 1 } { 2 } x }\).

1. Find the \(x\)-coordinate of \(M\), the maximum point of the curve.
2. Find the area of the shaded region enclosed by the curve, the \(x\)-axis and the line \(x = 1\), giving your answer in terms of e.

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

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.

9 The complex number \(u\) is given by $$u = \frac { 3 + \mathrm { i } } { 2 - \mathrm { i } }$$

1. Express \(u\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
2. Find the modulus and argument of \(u\).
3. Sketch an Argand diagram showing the point representing the complex number \(u\). Show on the same diagram the locus of the point representing the complex number \(z\) such that \(| z - u | = 1\).
4. Using your diagram, calculate the least value of \(| z |\) for points on this locus.

3 Use integration by parts to show that $$\int _ { 2 } ^ { 4 } \ln x \mathrm {~d} x = 6 \ln 2 - 2$$

3 The curve \(y = \frac { \mathrm { e } ^ { x } } { \cos x }\), for \(- \frac { 1 } { 2 } \pi < x < \frac { 1 } { 2 } \pi\), has one stationary point. Find the \(x\)-coordinate of this point.

4 A curve has equation \(y = \mathrm { e } ^ { - 3 x } \tan x\). Find the \(x\)-coordinates of the stationary points on the curve in the interval \(- \frac { 1 } { 2 } \pi < x < \frac { 1 } { 2 } \pi\). Give your answers correct to 3 decimal places.

9 \includegraphics[max width=\textwidth, alt={}, center]{8d134c65-af23-4508-acef-49b6ab49e374-3_504_910_625_614} The diagram shows the curve \(y = \frac { \ln x } { \sqrt { } x }\) and its maximum point \(M\). The curve cuts the \(x\)-axis at the point \(A\).

1. State the coordinates of \(A\).
2. Find the exact value of the \(x\)-coordinate of \(M\).
3. Using integration by parts, show that the area of the shaded region bounded by the curve, the \(x\)-axis and the line \(x = 4\) is equal to \(8 \ln 2 - 4\).

9 \includegraphics[max width=\textwidth, alt={}, center]{bbc19395-6f88-4a7c-b5d4-59ced9ccdcf2-4_597_895_258_625} The diagram shows the curve \(y = x ^ { 3 } \ln x\) and its minimum point \(M\).

1. Find the exact coordinates of \(M\).
2. Find the exact area of the shaded region bounded by the curve, the \(x\)-axis and the line \(x = 2\).

9 \includegraphics[max width=\textwidth, alt={}, center]{822f851a-7fae-43b8-9ebc-94588f01e51c-4_597_895_258_625} The diagram shows the curve \(y = x ^ { 3 } \ln x\) and its minimum point \(M\).

1. Find the exact coordinates of \(M\).
2. Find the exact area of the shaded region bounded by the curve, the \(x\)-axis and the line \(x = 2\).