Verify stationary point location

5 A curve has equation \(y = 2 x + \frac { 1 } { ( x - 1 ) ^ { 2 } }\). Verify that the curve has a stationary point at \(x = 2\) and determine its nature.

3

1. Given that \(y = \mathrm { e } ^ { - x } \sin 2 x\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Hence show that the curve \(y = \mathrm { e } ^ { - x } \sin 2 x\) has a stationary point when \(x = \frac { 1 } { 2 } \arctan 2\).

3 A curve is defined by the equation \(y = 2 x \ln ( 1 + x )\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence verify that the origin is a stationary point of the curve.
2. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\), and use this to verify that the origin is a minimum point.
3. Using the substitution \(u = 1 + x\), show that \(\int \frac { x ^ { 2 } } { 1 + x } \mathrm {~d} x = \int \left( u - 2 + \frac { 1 } { u } \right) \mathrm { d } u\). Hence evaluate \(\int _ { 0 } ^ { 1 } \frac { x ^ { 2 } } { 1 + x } \mathrm {~d} x\), giving your answer in an exact form.
4. Using integration by parts and your answer to part (iii), evaluate \(\int _ { 0 } ^ { 1 } 2 x \ln ( 1 + x ) \mathrm { d } x\).

6. \includegraphics[max width=\textwidth, alt={}, center]{0214eebf-93f2-4338-9222-443000115225-4_346_1040_303_548} \section*{Figure 1} Figure 1 shows a sketch of the curve \(C _ { 1 }\) with equation $$y = \cos ( \cos x ) \sin x \quad \text { for } \quad 0 \leqslant x \leqslant \pi$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
2. Hence verify that the turning point is at \(x = \frac { \pi } { 2 }\) and find the \(y\) coordinate of this point.
3. Find the area of the region bounded by \(C _ { 1 }\) and the positive \(x\)-axis between \(x = 0\) and \(x = \pi\) Figure 2 shows a sketch of the curve \(C _ { 1 }\) and the curve \(C _ { 2 }\) with equation $$y = \sin ( \cos x ) \sin x \quad \text { for } \quad 0 \leqslant x \leqslant \pi$$ \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{0214eebf-93f2-4338-9222-443000115225-4_519_1065_1631_484} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} The curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect at the origin and the point \(A ( a , b )\) ,where \(a < \pi\)
4. Find \(a\) and \(b\) ,giving \(b\) in a form not involving trigonometric functions.
5. Find the area of the shaded region between \(C _ { 1 }\) and \(C _ { 2 }\)

Show that the curve \(y = x^2 \ln x\) has a stationary point when \(x = \frac{1}{\sqrt{e}}\). [6]