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.

4 questions · Challenging +1.5

8.05d Partial differentiation: first and second order, mixed derivatives8.05f Nature of stationary points: classify using Hessian matrix
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OCR MEI FP3 2009 June Q2
24 marks Challenging +1.8
2 A surface has equation \(z = 3 x ( x + y ) ^ { 3 } - 2 x ^ { 3 } + 24 x\).
  1. Find \(\frac { \partial z } { \partial x }\) and \(\frac { \partial z } { \partial y }\).
  2. Find the coordinates of the three stationary points on the surface.
  3. Find the equation of the normal line at the point \(\mathrm { P } ( 1 , - 2,19 )\) on the surface.
  4. The point \(\mathrm { Q } ( 1 + k , - 2 + h , 19 + 3 h )\) is on the surface and is close to P . Find an approximate expression for \(k\) in terms of \(h\).
  5. Show that there is only one point on the surface at which the tangent plane has an equation of the form \(27 x - z = d\). Find the coordinates of this point and the corresponding value of \(d\).
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