Stationary points of surface (multivariable)

A question is this type if and only if it involves a surface z = f(x, y) and requires finding stationary points using partial derivatives with respect to both x and y simultaneously.

5 questions · Challenging +1.1

8.05d Partial differentiation: first and second order, mixed derivatives8.05f Nature of stationary points: classify using Hessian matrix
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OCR Further Additional Pure AS 2018 June Q2
9 marks Standard +0.8
2 The surface with equation \(z = 6 x ^ { 3 } + \frac { 1 } { 9 } y ^ { 2 } + x ^ { 2 } y\) has two stationary points.
  1. Verify that one of these stationary points is at the origin.
  2. Find the coordinates of the second stationary point.
OCR Further Additional Pure AS 2022 June Q2
6 marks Challenging +1.2
2 The surface \(S\) has equation \(z = x ^ { 3 } + y ^ { 3 } - 2 x ^ { 2 } - 5 y ^ { 2 } + 3 x y\).
It is given that \(S\) has two stationary points; one at the origin, \(O\), and the other at the point \(A\).
Determine the coordinates of \(A\).
OCR Further Additional Pure AS 2023 June Q3
6 marks Challenging +1.2
3 A surface has equation \(z = x ^ { 2 } y ^ { 2 } - 3 x y + 2 x + y\) for all real values of \(x\) and \(y\). Determine the coordinates of all stationary points of this surface.
OCR Further Additional Pure AS 2020 November Q2
11 marks Challenging +1.2
2 An open-topped rectangular box is to be manufactured with a fixed volume of \(1000 \mathrm {~cm} ^ { 3 }\). The dimensions of the base of the box are \(x \mathrm {~cm}\) by \(y \mathrm {~cm}\). The surface area of the box is \(A \mathrm {~cm} ^ { 2 }\).
  1. Show that \(\mathrm { A } = \mathrm { xy } + 2000 \left( \frac { 1 } { \mathrm { x } } + \frac { 1 } { \mathrm { y } } \right)\).
    1. Use partial differentiation to determine, in exact form, the values of \(x\) and \(y\) for which \(A\) has a stationary value.
    2. Find the stationary value of \(A\).
OCR MEI FP3 2015 June Q2
24 marks Challenging +1.2
2 A surface has equation \(z = 3 x ^ { 2 } - 12 x y + 2 y ^ { 3 } + 60\).
  1. Show that the point \(\mathrm { A } ( 8,4 , - 4 )\) is a stationary point on the surface. Find the coordinates of the other stationary point, B , on this surface.
  2. A point P with coordinates \(( 8 + h , 4 + k , p )\) lies on the surface.
    (A) Show that \(p = - 4 + 3 ( h - 2 k ) ^ { 2 } + 2 k ^ { 2 } ( 6 + k )\).
    (B) Deduce that the stationary point A is a local minimum.
    (C) By considering sections of the surface near to B in each of the planes \(x = 0\) and \(y = 0\), investigate the nature of the stationary point B .
  3. The point Q with coordinates \(( 1,1,53 )\) lies on the surface. Show that the equation of the tangent plane at Q is $$6 x + 6 y + z = 65$$
  4. The tangent plane at the point R has equation \(6 x + 6 y + z = \lambda\) where \(\lambda \neq 65\). Find the coordinates of R .