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Exercise 10

Avg difficulty 2.5

Consider the function f(x,y,z)=x4+y4+z4−4xyzf(x,y,z) = x^4 + y^4 +z^4 - 4xyz.

  1. Show that ff has a critical point at (−1,1,−1)(-1,1,-1) and construct the Hessian HH at that point.

  2. Find the eigenvalues of HH. Is this a definite matrix of some kind?

  3. What does this imply about whether (−1,1,−1)(-1,1,-1) is a local maximum or minimum?

Austin, Understanding Linear Algebra, §7.2 Quadratic forms, Exercise 10

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Adapted from Understanding Linear Algebra © 2023 David Austin. Licensed under CC BY 4.0. Source: Understanding Linear Algebra. Imported at revision 3e6dced.