Find stationary points

2. The curve \(C\) has equation $$x ^ { 2 } - 3 x y - 4 y ^ { 2 } + 64 = 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
2. Find the coordinates of the points on \(C\) where \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) (Solutions based entirely on graphical or numerical methods are not acceptable.)

1. The curve \(C\) has equation

$$x ^ { 2 } + x y + y ^ { 2 } - 4 x - 5 y + 1 = 0$$

1. Use implicit differentiation to find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
2. Find the \(x\) coordinates of the two points on \(C\) where \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) Give exact answers in their simplest form.
   (Solutions based entirely on graphical or numerical methods are not acceptable.)

11. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the closed curve with equation $$( x + y ) ^ { 3 } + 10 y ^ { 2 } = 108 x$$

1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 108 - 3 ( x + y ) ^ { 2 } } { 20 y + 3 ( x + y ) ^ { 2 } }$$ The curve is used to model the shape of a cycle track with both \(x\) and \(y\) measured in km .
   The points \(P\) and \(Q\) represent points that are furthest north and furthest south of the origin \(O\), as shown in Figure 4. Using the result given in part (a),
2. find how far the point \(Q\) is south of \(O\). Give your answer to the nearest 100 m .

3 A curve has equation \(2 y ^ { 2 } + y = 9 x ^ { 2 } + 1\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). Hence find the gradient of the curve at the point \(\mathrm { A } ( 1,2 )\).
2. Find the coordinates of the points on the curve at which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\).

5 Given that \(y ^ { 3 } = x y - x ^ { 2 }\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y - 2 x } { 3 y ^ { 2 } - x }\).
Hence show that the curve \(y ^ { 3 } = x y - x ^ { 2 }\) has a stationary point when \(x = \frac { 1 } { 8 }\).

2 Fig. 9 shows the curve with equation \(y ^ { 3 } = \frac { x ^ { 3 } } { 2 x - 1 }\). It has an asymptote \(x = a\) and turning point P . \begin{figure}[h] \end{figure}

1. Write down the value of \(a\).
2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 x ^ { 3 } - 3 x ^ { 2 } } { 3 y ^ { 2 } ( 2 x - 1 ) ^ { 2 } }\). Hence find the coordinates of the turning point P , giving the \(y\)-coordinate to 3 significant figures.
3. Show that the substitution \(u = 2 x - 1\) transforms \(\int \frac { x } { \sqrt [ 3 ] { 2 x - 1 } } \mathrm {~d} x\) to \(\frac { 1 } { 4 } \int \left( u ^ { \frac { 2 } { 3 } } + u ^ { - \frac { 1 } { 3 } } \right) \mathrm { d } u\). Hence find the exact area of the region enclosed by the curve \(y ^ { 3 } = \frac { x ^ { 3 } } { 2 x - 1 }\), the \(x\)-axis and the lines \(x = 1\) and \(x = 4.5\).

2 A curve has equation \(x ^ { 2 } + 2 y ^ { 2 } = 4 x\).

1. By differentiating implicitly, find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
   [0pt]
2. Hence find the exact coordinates of the stationary points of the curve. [You need not determine their nature.]

4 Fig. 7 shows the curve \(x ^ { 3 } + y ^ { 3 } = 3 x y\). The point P is a turning point of the curve. \begin{figure}[h] \end{figure}

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y - x ^ { 2 } } { y ^ { 2 } - x }\).
2. Hence find the exact \(x\)-coordinate of P .

4 A curve has equation \(2 y ^ { 2 } + y = 9 x ^ { 2 } + 1\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). Hence find the gradient of the curve at the point \(\mathrm { A } ( 1,2 )\).
2. Find the coordinates of the points on the curve at which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\).

1 Fig. 9 shows the curve with equation \(y ^ { 3 } = \frac { x ^ { 3 } } { 2 x - 1 }\). It has an asymptote \(x = a\) and turning point P . \begin{figure}[h] \end{figure}

1. Write down the value of \(a\).
2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 x ^ { 3 } - 3 x ^ { 2 } } { 3 y ^ { 2 } ( 2 x - 1 ) ^ { 2 } }\). Hence find the coordinates of the turning point P , giving the \(y\)-coordinate to 3 significant figures.
3. Show that the substitution \(u = 2 x - 1\) transforms \(\int \frac { x } { \sqrt [ 3 ] { 2 x - 1 } } \mathrm {~d} x\) to \(\frac { 1 } { 4 } \int \left( u ^ { \frac { 2 } { 3 } } + u ^ { - \frac { 1 } { 3 } } \right) \mathrm { d } u\). Hence find the exact area of the region enclosed by the curve \(y ^ { 3 } = \frac { x ^ { 3 } } { 2 x - 1 }\), the \(x\)-axis and the lines \(x = 1\) and \(x = 4.5\).

7 The equation of a curve is \(2 x ^ { 2 } + x y + y ^ { 2 } = 14\). Show that there are two stationary points on the curve and find their coordinates.

3 The equation of a curve is \(x ^ { 2 } y - x y ^ { 2 } = 2\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y ^ { 2 } - 2 x y } { x ^ { 2 } - 2 x y }\).
2. (a) Show that, if \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\), then \(y = 2 x\).
   (b) Hence find the coordinates of the point on the curve where the tangent is parallel to the \(x\)-axis.

5

1. For the curve \(2 x ^ { 2 } + x y + y ^ { 2 } = 14\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
2. Deduce that there are two points on the curve \(2 x ^ { 2 } + x y + y ^ { 2 } = 14\) at which the tangents are parallel to the \(x\)-axis, and find their coordinates.

2. A curve has the equation $$2 x ^ { 2 } + x y - y ^ { 2 } + 18 = 0$$ Find the coordinates of the points where the tangent to the curve is parallel to the \(x\)-axis.

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 \(B\).

1. Find the coordinates of the points \(A , B\) and \(C\).
2. Prove that \(B\) is a stationary point. Given that the region \(O C B\) is convex,
3. 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.
4. 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

7 Fig. 7 shows the curve \(x ^ { 3 } + y ^ { 3 } = 3 x y\). The point P is a turning point of the curve. \begin{figure}[h] \end{figure}

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y - x ^ { 2 } } { y ^ { 2 } - x }\).
2. Hence find the exact \(x\)-coordinate of P . Section B (36 marks)

9 Fig. 9 shows the curve with equation \(y ^ { 3 } = \frac { x ^ { 3 } } { 2 x - 1 }\). It has an asymptote \(x = a\) and turning point P . \begin{figure}[h] \end{figure}

1. Write down the value of \(a\).
2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 x ^ { 3 } - 3 x ^ { 2 } } { 3 y ^ { 2 } ( 2 x - 1 ) ^ { 2 } }\). Hence find the coordinates of the turning point P , giving the \(y\)-coordinate to 3 significant figures.
3. Show that the substitution \(u = 2 x - 1\) transforms \(\int \frac { x } { \sqrt [ 3 ] { 2 x - 1 } } \mathrm {~d} x\) to \(\frac { 1 } { 4 } \int \left( u ^ { \frac { 2 } { 3 } } + u ^ { - \frac { 1 } { 3 } } \right) \mathrm { d } u\). Hence find the exact area of the region enclosed by the curve \(y ^ { 3 } = \frac { x ^ { 3 } } { 2 x - 1 }\), the \(x\)-axis and the lines \(x = 1\) and \(x = 4.5\).

8 The equation of a curve is \(x ^ { 3 } + y ^ { 3 } = 6 x y\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
2. Show that the point \(\left( 2 ^ { \frac { 4 } { 3 } } , 2 ^ { \frac { 5 } { 3 } } \right)\) lies on the curve and that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) at this point.
3. The point \(( a , a )\), where \(a > 0\), lies on the curve. Find the value of \(a\) and the gradient of the curve at this point.

3 The equation of a curve is \(x y ^ { 2 } = x ^ { 2 } + 1\). Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\), and hence find the coordinates of the stationary points on the curve.

5 Find the coordinates of the two stationary points on the curve with equation $$x ^ { 2 } + 4 x y + 2 y ^ { 2 } + 18 = 0$$

7 The curve \(C\) has equation \(x ^ { 2 } + 2 x y - 4 y ^ { 2 } + 20 = 0\). Show that if the tangent to \(C\) at the point \(( x , y )\) is parallel to the \(x\)-axis then \(x + y = 0\). Hence find the coordinates of the stationary points on \(C\), and determine their nature.

10 The curve \(C\) has equation \(x ^ { 3 } + y ^ { 3 } = 3 x y\), for \(x > 0\) and \(y > 0\). Find a relationship between \(x\) and \(y\) when \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\). Find the exact coordinates of the turning point of \(C\), and determine the nature of this turning point.

1. A curve \(C\) is given by the equation

$$\sin x + \cos y = 0.5 \quad - \frac { \pi } { 2 } \leqslant x < \frac { 3 \pi } { 2 } , - \pi < y < \pi$$ A point \(P\) lies on \(C\).
The tangent to \(C\) at the point \(P\) is parallel to the \(x\)-axis.
Find the exact coordinates of all possible points \(P\), justifying your answer.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

13 Determine the coordinates of the turning points on the curve with equation $$y ^ { 2 } + x y + x ^ { 2 } - x = 1 .$$

12 Fig. 12 shows the curve \(2 x ^ { 3 } + y ^ { 3 } = 5 y\). \includegraphics[max width=\textwidth, alt={}, center]{e9f3a5f3-210b-453d-9ff5-8518340f5689-08_841_606_900_212}

1. Find the gradient of the curve \(2 x ^ { 3 } + y ^ { 3 } = 5 y\) at the point \(( 1,2 )\), giving your answer in exact form.
2. Show that all the stationary points of the curve lie on the \(y\)-axis.