Find stationary points

5 A curve is defined implicitly by the equation \(2 x ^ { 2 } + 3 x y + y ^ { 2 } + 2 = 0\).

1. Show that \(\frac { d y } { d x } = - \frac { 4 x + 3 y } { 3 x + 2 y }\).
2. In this question you must show detailed reasoning. Find the coordinates of the stationary points of the curve.

6 The surface \(C\) is given by the equation \(z = x ^ { 2 } + y ^ { 3 } + a x y\) for all real \(x\) and \(y\), where \(a\) is a non-zero real number.

1. Show that \(C\) has two stationary points, one of which is at the origin, and give the coordinates of the second in terms of \(a\).
2. Determine the nature of these stationary points of \(C\).
3. Explain what can be said about the location and nature of the stationary point(s) of the surface given by the equation \(z = x ^ { 2 } + y ^ { 3 }\) for all real \(x\) and \(y\).

6 A curve is defined by the equation \(9 x ^ { 2 } - 6 x y + 4 y ^ { 2 } = 3\). Find the coordinates of the two stationary points of this curve.

1. 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.

6 A surface \(S\) has equation \(z = \mathrm { f } ( x , y )\), where \(\mathrm { f } ( x , y ) = 2 x ^ { 2 } - y ^ { 2 } + 3 x y + 17 y\). It is given that \(S\) has a single stationary point, \(P\).

1. Determine the coordinates, and the nature, of \(P\).
2. Find the equation of the tangent plane to \(S\) at the point \(Q ( 1,2,38 )\).

5 A curve is defined by the equation $$y ^ { 2 } - x y + 3 x ^ { 2 } - 5 = 0$$

1. Find the \(y\)-coordinates of the two points on the curve where \(x = 1\).
2. 1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y - 6 x } { 2 y - x }\).
   2. Find the gradient of the curve at each of the points where \(x = 1\).
   3. Show that, at the two stationary points on the curve, \(33 x ^ { 2 } - 5 = 0\).

13 In this question you must show detailed reasoning. Find the exact values of the \(x\)-coordinates of the stationary points of the curve \(x ^ { 3 } + y ^ { 3 } = 3 x y + 35\).

15 The curve with equation $$x ^ { 2 } + 2 y ^ { 3 } - 4 x y = 0$$ has a single stationary point at \(P\) as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{6a03a035-ff32-4734-864b-a076aa9cbec0-26_656_1138_548_450} 15

1. Show that the \(y\)-coordinate of \(P\) satisfies the equation $$y ^ { 2 } ( y - 2 ) = 0$$ 15
2. Hence, find the coordinates of \(P\) [0pt] [2 marks]

12 A sculpture formed from a prism is fixed on a horizontal platform, as shown in the diagram. The shape of the cross-section of the sculpture can be modelled by the equation \(x ^ { 2 } + 2 x y + 2 y ^ { 2 } = 10\), where \(x\) and \(y\) are measured in metres. The \(x\) and \(y\) axes are horizontal and vertical respectively. \includegraphics[max width=\textwidth, alt={}, center]{66b1c52c-0475-493a-8c0a-abf4c2c7b07d-18_214_451_776_858} Find the maximum vertical height above the platform of the sculpture.

6 Find the coordinates of the stationary point of the curve with equation $$( x + y - 2 ) ^ { 2 } = \mathrm { e } ^ { y } - 1$$ \(7 \quad\) A function f has domain \(\mathbb { R }\) and range \(\{ y \in \mathbb { R } : y \geq \mathrm { e } \}\) The graph of \(y = \mathrm { f } ( x )\) is shown. \includegraphics[max width=\textwidth, alt={}, center]{5e475c96-6bbb-44fb-a411-706dac20e2d6-08_922_1108_447_466} The gradient of the curve at the point \(( x , y )\) is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = ( x - 1 ) \mathrm { e } ^ { x }\) Find an expression for \(\mathrm { f } ( x )\).
Fully justify your answer.