Find stationary points - logarithmic functions

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\includegraphics[max width=\textwidth, alt={}, center]{567c3d72-c633-4ae0-8605-f63f93d718c4-18_979_679_262_731} The diagram shows part of the curve \(y = 1 - \frac { 4 } { ( 2 x + 1 ) ^ { 2 } }\). The curve intersects the \(x\)-axis at \(A\). The normal to the curve at \(A\) intersects the \(y\)-axis at \(B\).

1. Obtain expressions for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\int y \mathrm {~d} x\).
2. Find the coordinates of \(B\).
3. Find, showing all necessary working, the area of the shaded region.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

6 The curve \(y = 4 x ^ { 2 } \ln x\) has one stationary point.

1. Find the coordinates of this stationary point, giving your answers correct to 3 decimal places.
2. Determine whether this point is a maximum or a minimum point.

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.

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

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.

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\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 \(8 \ln 2 - 4\).

5 The curve with equation \(y = x ^ { 2 } \ln x\), where \(x > 0\), has one stationary point.

1. Find the \(x\)-coordinate of this point, giving your answer in terms of e .
2. Determine whether this point is a maximum or a minimum point.

6 The curve with equation \(y = x \ln x\) has one stationary point.

1. Find the exact coordinates of this point, giving your answers in terms of e .
2. Determine whether this point is a maximum or a minimum point.

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\includegraphics[max width=\textwidth, alt={}, center]{e814d76c-8757-4cc4-a69c-e3636b4cab16-3_611_1084_648_532} The diagram shows the curve \(y = \frac { \ln x } { x ^ { 2 } }\) and its maximum point \(M\).

1. Find the exact coordinates of \(M\).
2. Use the trapezium rule with three intervals to estimate the value of $$\int _ { 1 } ^ { 4 } \frac { \ln x } { x ^ { 2 } } \mathrm {~d} x$$ giving your answer correct to 2 decimal places.

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1. Find the exact value of the \(x\)-coordinate of the stationary point of the curve \(y = x \ln x\).
2. The equation of a curve is \(y = \frac { 4 x + c } { 4 x - c }\), where \(c\) is a non-zero constant. Show by differentiation that this curve has no stationary points.

7 Fig. 7 shows the curve $$y = 2 x - x \ln x , \text { where } x > 0 .$$ The curve crosses the \(x\)-axis at A , and has a turning point at B . The point C on the curve has \(x\)-coordinate 1 . Lines CD and BE are drawn parallel to the \(y\)-axis. \begin{figure}[h] \end{figure}

1. Find the \(x\)-coordinate of A , giving your answer in terms of e .
2. Find the exact coordinates of B .
3. Show that the tangents at A and C are perpendicular to each other.
4. Using integration by parts, show that $$\int x \ln x \mathrm {~d} x = \frac { 1 } { 2 } x ^ { 2 } \ln x - \frac { 1 } { 4 } x ^ { 2 } + c$$ Hence find the exact area of the region enclosed by the curve, the \(x\)-axis and the lines CD and BE . \section*{[Question 8 is printed overleaf.]}

8 Fig. 8 shows the curve \(y = 3 \ln x + x - x ^ { 2 }\).
The curve crosses the \(x\)-axis at P and Q , and has a turning point at R . The \(x\)-coordinate of Q is approximately 2.05 . \begin{figure}[h] \end{figure}

1. Verify that the coordinates of P are \(( 1,0 )\).
2. Find the coordinates of R , giving the \(y\)-coordinate correct to 3 significant figures. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\), and use this to verify that R is a maximum point.
3. Find \(\int \ln x \mathrm {~d} x\). Hence calculate the area of the region enclosed by the curve and the \(x\)-axis between P and Q , giving your answer to 2 significant figures.

6 The function \(\mathrm { f } ( x )\) is defined as \(\mathrm { f } ( x ) = \frac { \ln x } { x }\). The graph of the function is shown in Fig. 6 . \begin{figure}[h] \end{figure}

1. Give the coordinates of the point, P , where the curve crosses the \(x\)-axis.
2. Use calculus to find the coordinates of the stationary point, Q , and show that it is a maximum.

6. The curve \(C\) has the equation \(y = x ^ { 2 } - 5 x + 2 \ln \frac { x } { 3 } , x > 0\).

1. Show that the normal to \(C\) at the point where \(x = 3\) has the equation $$3 x + 5 y + 21 = 0$$
2. Find the \(x\)-coordinates of the stationary points of \(C\).

6 The function \(\mathrm { f } ( x ) = \frac { \sin x } { 2 - \cos x }\) has domain \(- \pi \leqslant x \leqslant \pi\).
Fig. 8 shows the graph of \(y = \mathrm { f } ( x )\) for \(0 \leqslant x \leqslant \pi\). \begin{figure}[h] \end{figure}

1. Find \(\mathrm { f } ( - x )\) in terms of \(\mathrm { f } ( x )\). Hence sketch the graph of \(y = \mathrm { f } ( x )\) for the complete domain \(- \pi \leqslant x \leqslant \pi\).
2. Show that \(\mathrm { f } ^ { \prime } ( x ) = \frac { 2 \cos x - 1 } { ( 2 - \cos x ) ^ { 2 } }\). Hence find the exact coordinates of the turning point P . State the range of the function \(\mathrm { f } ( x )\), giving your answer exactly.
3. Using the substitution \(u = 2 - \cos x\) or otherwise, find the exact value of \(\int _ { 0 } ^ { \pi } \frac { \sin x } { 2 \cos x } \mathrm {~d} x\).
4. Sketch the graph of \(y = \mathrm { f } ( 2 x )\).
5. Using your answers to parts (iii) and (iv), write down the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \frac { \sin 2 x } { 2 \cos 2 x } \mathrm {~d} x\).

1 Fig. 8 shows the curve \(y = 3 \ln x + x - x ^ { 2 }\).
The curve crosses the \(x\)-axis at P and Q , and has a turning point at R . The \(x\)-coordinate of Q is approximately 2.05 . \begin{figure}[h] \end{figure}

1. Verify that the coordinates of P are \(( 1,0 )\).
2. Find the coordinates of R , giving the \(y\)-coordinate correct to 3 significant figures. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\), and use this to verify that R is a maximum point.
3. Find \(\int \ln x \mathrm {~d} x\). Hence calculate the area of the region enclosed by the curve and the \(x\)-axis between P and Q , giving your answer to 2 significant figures.

2 Fig. 8 shows the curve \(y = 3 \ln x + x - x ^ { 2 }\).
The curve crosses the \(x\)-axis at P and Q , and has a turning point at R . The \(x\)-coordinate of Q is approximately 2.05 . \begin{figure}[h] \end{figure}

1. Verify that the coordinates of P are \(( 1,0 )\).
2. Find the coordinates of R , giving the \(y\)-coordinate correct to 3 significant figures. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\), and use this to verify that R is a maximum point.
3. Find \(\int \ln x \mathrm {~d} x\). Hence calculate the area of the region enclosed by the curve and the \(x\)-axis between P and Q , giving your answer to 2 significant figures.

1 Fig. 7 shows the curve $$y = 2 x - x \ln x , \text { where } x > 0 .$$ The curve crosses the \(x\)-axis at A , and has a turning point at B . The point C on the curve has \(x\)-coordinate 1 . Lines CD and BE are drawn parallel to the \(y\)-axis. \begin{figure}[h] \end{figure}

1. Find the \(x\)-coordinate of A , giving your answer in terms of e .
2. Find the exact coordinates of B .
3. Show that the tangents at A and C are perpendicular to each other.
4. Using integration by parts, show that $$\int x \ln x \mathrm {~d} x = \frac { 1 } { 2 } x ^ { 2 } \ln x - \frac { 1 } { 4 } x ^ { 2 } + c$$ Hence find the exact area of the region enclosed by the curve, the \(x\)-axis and the lines CD and BE .

1. $$\mathrm { f } ( x ) = x ^ { \left( x ^ { 2 } \right) } \quad x > 0$$ Use logarithms to find the \(x\) coordinate of the stationary point of the curve with equation \(y = \mathrm { f } ( x )\) ．