Finding maximum/minimum on curve

11
\includegraphics[max width=\textwidth, alt={}, center]{ed5b77ae-6eac-4e73-bc43-613433abd3e1-16_723_942_260_598} The diagram shows part of the curve \(y = \sqrt { } ( 4 x + 1 ) + \frac { 9 } { \sqrt { } ( 4 x + 1 ) }\) and the minimum point \(M\).

1. Find expressions for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\int y \mathrm {~d} x\).
2. Find the coordinates of \(M\).
   The shaded region is bounded by the curve, the \(y\)-axis and the line through \(M\) parallel to the \(x\)-axis.
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.

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.

10
\includegraphics[max width=\textwidth, alt={}, center]{5d76d7cb-c2cb-4fa4-8133-d5d43702b293-3_366_764_1914_687} The diagram shows the curve \(y = \frac { x ^ { 2 } } { 1 + x ^ { 3 } }\) for \(x \geqslant 0\), and its maximum point \(M\). The shaded region \(R\) is enclosed by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = p\).

1. Find the exact value of the \(x\)-coordinate of \(M\).
2. Calculate the value of \(p\) for which the area of \(R\) is equal to 1 . Give your answer correct to 3 significant figures.

10
\includegraphics[max width=\textwidth, alt={}, center]{efa44efb-9350-4d54-b5e1-4f722781a5f3-3_366_764_1914_687} The diagram shows the curve \(y = \frac { x ^ { 2 } } { 1 + x ^ { 3 } }\) for \(x \geqslant 0\), and its maximum point \(M\). The shaded region \(R\) is enclosed by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = p\).

1. Find the exact value of the \(x\)-coordinate of \(M\).
2. Calculate the value of \(p\) for which the area of \(R\) is equal to 1 . Give your answer correct to 3 significant figures.

9
\includegraphics[max width=\textwidth, alt={}, center]{ce3c4a9c-bf83-4d28-96e2-ef31c3673dea-12_375_645_274_742} The diagram shows the curve \(y = x \mathrm { e } ^ { - \frac { 1 } { 4 } x ^ { 2 } }\), for \(x \geqslant 0\), and its maximum point \(M\).

1. Find the exact coordinates of \(M\).
2. Using the substitution \(x = \sqrt { u }\), or otherwise, find by integration the exact area of the shaded region bounded by the curve, the \(x\)-axis and the line \(x = 3\).

7 Fig. 7 shows the curve \(y = \frac { x ^ { 2 } } { 1 + 2 x ^ { 3 } }\). It is undefined at \(x = a\); the line \(x = a\) is a vertical asymptote. \begin{figure}[h] \end{figure}

1. Calculate the value of \(a\), giving your answer correct to 3 significant figures.
2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 x - 2 x ^ { 4 } } { \left( 1 + 2 x ^ { 3 } \right) ^ { 2 } }\). Hence determine the coordinates of the turning points of the curve.
3. Show that the area of the region between the curve and the \(x\)-axis from \(x = 0\) to \(x = 1\) is \(\frac { 1 } { 6 } \ln 3\).

3 Fig. 7 shows the curve \(y = \frac { x ^ { 2 } } { 1 + 2 x ^ { 3 } }\). It is undefined at \(x = a\); the line \(x = a\) is a vertical asymptote. \begin{figure}[h] \end{figure}

1. Calculate the value of \(a\), giving your answer correct to 3 significant figures.
2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 x - 2 x ^ { 4 } } { \left( 1 + 2 x ^ { 3 } \right) ^ { 2 } }\). Hence determine the coordinates of the turning points of the curve.
3. Show that the area of the region between the curve and the \(x\)-axis from \(x = 0\) to \(x = 1\) is \(\frac { 1 } { 6 } \ln 3\).