Finding stationary points on surfaces

A question is this type if and only if it asks to find coordinates of stationary points by setting partial derivatives to zero.

3 questions · Challenging +1.4

Sort by: Default | Easiest first | Hardest first
OCR Further Additional Pure 2022 June Q9
9 marks Challenging +1.2
9 For all real values of \(x\) and \(y\) the surface \(S\) has equation \(z = 4 x ^ { 2 } + 4 x y + y ^ { 2 } + 6 x + 3 y + k\), where \(k\) is a constant and an integer.
  1. Find \(\frac { \partial z } { \partial x }\) and \(\frac { \partial z } { \partial y }\).
  2. Determine the smallest value of the integer \(k\) for which the whole of \(S\) lies above the \(x - y\) plane.
OCR Further Additional Pure 2021 November Q5
8 marks Challenging +1.8
5 The surface \(S\) has equation \(x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = x y z - 1\).
  1. Show that \(( 2 z - x y ) \left( x \frac { \partial z } { \partial x } + y \frac { \partial z } { \partial y } \right) = 2 \left( 1 + z ^ { 2 } \right)\).
  2. Deduce that \(S\) has no stationary point.
OCR Further Additional Pure 2018 December Q7
11 marks Challenging +1.2
7 For each value of \(t\), the surface \(S _ { t }\) has equation \(z = t x ^ { 2 } + y ^ { 2 } + 3 x y - y\).
  1. Verify that there are no stationary points on \(S _ { t }\) when \(t = \frac { 9 } { 4 }\).
  2. Determine, as \(t\) varies, the nature of any stationary point(s) of \(S _ { t }\).
    (You do not have to find the coordinates of the stationary points.) \section*{OCR} Oxford Cambridge and RSA