Find stationary points

\includegraphics{figure_5} The diagram shows the curve with equation $$x^3 + xy^2 + ay^2 - 3ax^2 = 0,$$ where \(a\) is a positive constant. The maximum point on the curve is \(M\). Find the \(x\)-coordinate of \(M\) in terms of \(a\). [6]

A curve has equation $$x^2 + 2xy - 3y^2 + 16 = 0.$$ Find the coordinates of the points on the curve where \(\frac{dy}{dx} = 0\). [7]

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

1. Find \(\frac{dy}{dx}\) in terms of \(x\) and \(y\). [5]
2. Find the coordinates of the points on \(C\) where \(\frac{dy}{dx} = 0\) (Solutions based entirely on graphical or numerical methods are not acceptable.) [6]

Fig. 7 shows the curve \(x^3 + y^3 = 3xy\). The point P is a turning point of the curve. \includegraphics{figure_7}

1. Show that \(\frac{dy}{dx} = \frac{y - x^2}{y^2 - x}\). [4]
2. Hence find the exact \(x\)-coordinate of P. [4]

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

1. By differentiating implicitly, find \(\frac{dy}{dx}\) in terms of \(x\) and \(y\). [3]
2. Hence find the exact coordinates of the stationary points of the curve. [You need not determine their nature.] [3]

The equation of a curve is \(2x^2 + xy + y^2 = 14\). Show that there are two stationary points on the curve and find their coordinates. [8]

Find the coordinates of the stationary points of the curve with equation $$x^3 + y^3 - 3xy = 48$$ and determine their nature. [14]

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 \(x^2 + 2xy + 2y^2 = 10\), where \(x\) and \(y\) are measured in metres. The \(x\) and \(y\) axes are horizontal and vertical respectively. \includegraphics{figure_12} Find the maximum vertical height above the platform of the sculpture. [8 marks]

Find the coordinates of the stationary point of the curve with equation \((x + y - 2)^2 = e^y - 1\) [7 marks]

Fig. 12 shows the curve \(2x^3 + y^3 = 5y\). \includegraphics{figure_12}

1. Find the gradient of the curve \(2x^3 + y^3 = 5y\) at the point \((1,2)\), giving your answer in exact form. [4]
2. Show that all the stationary points of the curve lie on the \(y\)-axis. [2]

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 \(x^2 + 2xy + 2y^2 = 10\), where \(x\) and \(y\) are measured in metres. The \(x\) and \(y\) axes are horizontal and vertical respectively. \includegraphics{figure_3} Find the maximum vertical height above the platform of the sculpture. [8 marks]

A curve \(C\) is given by the equation $$\sin x + \cos y = 0.5 \quad -\frac{\pi}{2} \leq x < \frac{3\pi}{2}, -\pi < y < \pi$$ 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.) [5]

A curve \(C\) has equation $$x^3 + y^3 = 3xy + 48$$ Prove that \(C\) has two stationary points and find their coordinates. [7]

In this question you must show detailed reasoning. Find the exact values of the x-coordinates of the stationary points of the curve \(x^3 + y^3 = 3xy + 35\). [9]

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

1. Determine the coordinates, and the nature, of \(P\). [8]
2. Find the equation of the tangent plane to \(S\) at the point \(Q(1, 2, 38)\). [2]