5 It is given that \(\int _ { 1 } ^ { a } \left( \frac { 4 } { 1 + 2 x } + \frac { 3 } { x } \right) \mathrm { d } x = \ln 10\), where \(a\) is a constant greater than 1 .
- Show that \(a = \sqrt [ 3 ] { 90 ( 1 + 2 a ) ^ { - 2 } }\).
- Use an iterative formula, based on the equation in (a), to find the value of \(a\) correct to 3 significant figures. Use an initial value of 1.7 and give the result of each iteration to 5 significant figures.
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The diagram shows the curve with equation \(y = \frac { 4 \mathrm { e } ^ { 2 x } + 9 } { \mathrm { e } ^ { x } + 2 }\). The curve has a minimum point \(M\) and crosses the \(y\)-axis at the point \(P\). - Find the exact value of the gradient of the curve at \(P\).
- Find the exact coordinates of \(M\).