Classifying stationary points on surfaces

A question is this type if and only if it asks to determine the nature (maximum, minimum, or saddle point) of a stationary point using the Hessian or second derivative test.

4 questions · Challenging +1.4

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 2013 June Q2
24 marks Challenging +1.8
2 A surface has equation \(z = 2 \left( x ^ { 3 } + y ^ { 3 } \right) + 3 \left( x ^ { 2 } + y ^ { 2 } \right) + 12 x y\).
  1. For a point on the surface at which \(\frac { \partial z } { \partial x } = \frac { \partial z } { \partial y }\), show that either \(y = x\) or \(y = 1 - x\).
  2. Show that there are exactly two stationary points on the surface, and find their coordinates.
  3. The point \(\mathrm { P } \left( \frac { 1 } { 2 } , \frac { 1 } { 2 } , 5 \right)\) is on the surface, and \(\mathrm { Q } \left( \frac { 1 } { 2 } + h , \frac { 1 } { 2 } + h , 5 + w \right)\) is a point on the surface close to P . Find an approximate expression for \(h\) in terms of \(w\).
  4. Find the four points on the surface at which the normal line is parallel to the vector \(24 \mathbf { i } + 24 \mathbf { j } - \mathbf { k }\).
OCR Further Additional Pure 2019 June Q2
11 marks Standard +0.8
2 A surface has equation \(z = \mathrm { f } ( x , y )\) where \(\mathrm { f } ( x , y ) = x ^ { 2 } \sin y + 2 y \cos x\).
  1. Determine \(\mathrm { f } _ { x } , \mathrm { f } _ { y } , \mathrm { f } _ { x x } , \mathrm { f } _ { y y } , \mathrm { f } _ { x y }\) and \(\mathrm { f } _ { y x }\).
    1. Verify that \(z\) has a stationary point at \(\left( \frac { 1 } { 2 } \pi , \frac { 1 } { 2 } \pi , \frac { 1 } { 4 } \pi ^ { 2 } \right)\).
    2. Determine the nature of this stationary point.
OCR Further Additional Pure 2023 June Q6
11 marks Challenging +1.8
6 The surface \(S\) has equation \(z = x \sin y + \frac { y } { x }\) for \(x > 0\) and \(0 < y < \pi\).
  1. Determine, as a function of \(x\) and \(y\), the determinant of \(\mathbf { H }\), the Hessian matrix of \(S\).
  2. Given that \(S\) has just one stationary point, \(P\), use the answer to part (a) to deduce the nature of \(P\).
  3. The coordinates of \(P\) are \(( \alpha , \beta , \gamma )\). Show that \(\beta\) satisfies the equation \(\beta + \tan \beta = 0\).
OCR Further Additional Pure Specimen Q6
10 marks Challenging +1.2
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. (a) Determine the coordinates of \(P\).
    (b) Determine the nature of \(P\).
  2. Find the equation of the tangent plane to \(S\) at the point \(Q ( 1,2,38 )\).