Find stationary points

5 The equation of a curve is \(3 x ^ { 2 } + 2 x y + y ^ { 2 } = 6\). It is given that there are two points on the curve where the tangent is parallel to the \(x\)-axis.

1. Show by differentiation that, at these points, \(y = - 3 x\).
2. Hence find the coordinates of the two points.

7 The equation of a curve is $$x ^ { 2 } + y ^ { 2 } - 4 x y + 3 = 0$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 y - x } { y - 2 x }\).
2. Find the coordinates of each of the points on the curve where the tangent is parallel to the \(x\)-axis.

6 The equation of a curve is \(x y ( x + y ) = 2 a ^ { 3 }\), where \(a\) is a non-zero constant. Show that there is only one point on the curve at which the tangent is parallel to the \(x\)-axis, and find the coordinates of this point.

5 \includegraphics[max width=\textwidth, alt={}, center]{7c125770-1ded-4763-8453-b07ef43e83e9-2_446_601_1969_772} The diagram shows the curve with equation $$x ^ { 3 } + x y ^ { 2 } + a y ^ { 2 } - 3 a x ^ { 2 } = 0$$ where \(a\) is a positive constant. The maximum point on the curve is \(M\). Find the \(x\)-coordinate of \(M\) in terms of \(a\).

1. By differentiating \(\frac { 1 } { \cos x }\), show that the derivative of \(\sec x\) is \(\sec x \tan x\). Hence show that if \(y = \ln ( \sec x + \tan x )\) then \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sec x\).
2. Using the substitution \(x = ( \sqrt { } 3 ) \tan \theta\), find the exact value of $$\int _ { 1 } ^ { 3 } \frac { 1 } { \sqrt { \left( 3 + x ^ { 2 } \right) } } \mathrm { d } x$$ expressing your answer as a single logarithm.

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

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x ^ { 2 } - 2 x y } { x ^ { 2 } - y ^ { 2 } }\).
2. Find the coordinates of the points on the curve where the tangent is parallel to the \(x\)-axis.

6 A curve has equation $$\sin y \ln x = x - 2 \sin y$$ for \(- \frac { 1 } { 2 } \pi \leqslant y \leqslant \frac { 1 } { 2 } \pi\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
2. Hence find the exact \(x\)-coordinate of the point on the curve at which the tangent is parallel to the \(x\)-axis.

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

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y - x ^ { 2 } } { 2 y ^ { 2 } - x }\).
2. Find the coordinates of the point, other than the origin, where the curve has a tangent which is parallel to the \(x\)-axis.

4 The equation of a curve is \(x y ( x - 6 y ) = 9 a ^ { 3 }\), where \(a\) is a non-zero constant. Show that there is only one point on the curve at which the tangent is parallel to the \(x\)-axis, and find the coordinates of this point.

5 The equation of a curve is \(2 x ^ { 4 } + x y ^ { 3 } + y ^ { 4 } = 10\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 8 x ^ { 3 } + y ^ { 3 } } { 3 x y ^ { 2 } + 4 y ^ { 3 } }\).
2. Hence show that there are two points on the curve at which the tangent is parallel to the \(x\)-axis and find the coordinates of these points.

6 The equation of a curve is \(x ^ { 3 } y - 3 x y ^ { 3 } = 2 a ^ { 4 }\), where \(a\) is a non-zero constant.

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 x ^ { 2 } y - 3 y ^ { 3 } } { 9 x y ^ { 2 } - x ^ { 3 } }\).
2. Hence show that there are only two points on the curve at which the tangent is parallel to the \(x\)-axis and find the coordinates of these points.

5 The equation of a curve is \(2 x ^ { 2 } y - x y ^ { 2 } = a ^ { 3 }\), where \(a\) is a positive constant. Show that there is only one point on the curve at which the tangent is parallel to the \(x\)-axis and find the \(y\)-coordinate of this point.

7 The equation of a curve is \(2 x ^ { 3 } + y ^ { 3 } = 24\).

1. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\), and show that the gradient of the curve is never positive.
2. Find the coordinates of the two points on the curve at which the gradient is - 2 .

7 The equation of a curve is $$2 x ^ { 2 } + 3 y ^ { 2 } - 2 x y = 10 .$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y - 2 x } { 3 y - x }\).
2. Find the coordinates of the points on the curve where the tangent is parallel to the \(x\)-axis.

7 The equation of a curve is $$3 x ^ { 2 } - 4 x y + 2 y ^ { 2 } - 6 = 0$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 x - 2 y } { 2 x - 2 y }\).
2. Find the coordinates of each of the points on the curve where the tangent is parallel to the \(x\)-axis.

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

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x ^ { 2 } + 2 x y } { y ^ { 2 } - x ^ { 2 } }\).
2. Find the coordinates of the points on the curve where the tangent is parallel to the \(x\)-axis.

7 The equation of a curve is \(\ln ( x + y ) = x - 2 y\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x + y - 1 } { 2 ( x + y ) + 1 }\).
2. Find the coordinates of the point on the curve where the tangent is parallel to the \(x\)-axis. \(\quad\) [3]

7 The equation of a curve is \(x ^ { 3 } + y ^ { 2 } + 3 x ^ { 2 } + 3 y = 4\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 3 x ^ { 2 } + 6 x } { 2 y + 3 }\).
2. Hence find the coordinates of the points on the curve at which the tangent is parallel to the \(x\)-axis. [5]

1. The curve \(C\) has equation

$$81 y ^ { 3 } + 64 x ^ { 2 } y + 256 x = 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
2. Hence find the coordinates of the points on \(C\) where \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\)

2. The curve \(C\) has equation $$y ^ { 3 } + x ^ { 2 } y - 6 x = 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
2. Hence find the exact coordinates of the points on \(C\) for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\)

6. The curve \(C\) has equation $$16 y ^ { 3 } + 9 x ^ { 2 } y - 54 x = 0$$

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 \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\)

3. \begin{figure}[h] \end{figure} The curve \(C\), shown in Figure 1, has equation $$y ^ { 2 } x + 3 y = 4 x ^ { 2 } + k \quad y > 0$$ where \(k\) is a constant.

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\) The point \(P ( p , 2 )\), where \(p\) is a constant, lies on \(C\).
   Given that \(P\) is the minimum turning point on \(C\),
2. find
   1. the value of \(p\)
   2. the value of \(k\)

5. A set of curves is given by the equation \(\sin x + \cos y = 0.5\).

1. Use implicit differentiation to find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\). For \(- \pi < x < \pi\) and \(- \pi < y < \pi\),
2. find the coordinates of the points where \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\).

2. A curve has equation $$x ^ { 2 } + 2 x y - 3 y ^ { 2 } + 16 = 0 .$$ Find the coordinates of the points on the curve where \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\).

1. The curve \(C\) has equation

$$16 y ^ { 3 } + 9 x ^ { 2 } y - 54 x = 0$$

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 \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\).

3. $$x ^ { 2 } + y ^ { 2 } + 10 x + 2 y - 4 x y = 10$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\), fully simplifying your answer.
2. Find the values of \(y\) for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\)