4．Find the coordinates of the stationary points of the curve with equation
$$x ^ { 3 } + y ^ { 3 } - 3 x y = 48$$
and determine their nature．
\includegraphics[max width=\textwidth, alt={}, center]{7f1bc552-3850-43c5-b435-abc87b264f0a-3_553_749_401_618}
Figure 1 shows a sketch of part of the curve with equation
$$y = \sin ( \cos x ) .$$
The curve cuts the \(x\)-axis at the points \(A\) and \(C\) and the \(y\)-axis at the point \(B\).
- Find the coordinates of the points \(A , B\) and \(C\).
- Prove that \(B\) is a stationary point.
Given that the region \(O C B\) is convex,
- show that, for \(0 \leq x \leq \frac { \pi } { 2 }\),
$$\sin ( \cos x ) \leq \cos x$$
and
$$\left( 1 - \frac { 2 } { \pi } x \right) \sin 1 \leq \sin ( \cos x )$$
and state in each case the value or values of \(x\) for which equality is achieved.
- Hence show that
$$\frac { \pi } { 4 } \sin 1 < \int _ { 0 } ^ { \frac { \pi } { 2 } } \sin ( \cos x ) d x < 1$$
\includegraphics[max width=\textwidth, alt={}]{7f1bc552-3850-43c5-b435-abc87b264f0a-4_682_824_399_704}
Figure 2 shows a sketch of part of two curves \(C _ { 1 }\) and \(C _ { 2 }\) for \(y \geq 0\).
The equation of \(C _ { 1 }\) is \(y = m _ { 1 } - x ^ { n _ { 1 } }\) and the equation of \(C _ { 2 }\) is \(y = m _ { 2 } - x ^ { n _ { 2 } }\), where \(m _ { 1 }\), \(m _ { 2 } , n _ { 1 }\) and \(n _ { 2 }\) are positive integers with \(m _ { 2 } > m _ { 1 }\).
Both \(C _ { 1 }\) and \(C _ { 2 }\) are symmetric about the line \(x = 0\) and they both pass through the points \(( 3,0 )\) and \(( - 3,0 )\).
Given that \(n _ { 1 } + n _ { 2 } = 12\), find