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Mathematics · University

Hessian determinant of a quadratic

A positive determinant alone does not distinguish a local minimum from a maximum; inspect fₓₓ as well.

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Method and conditions

A positive determinant alone does not distinguish a local minimum from a maximum; inspect fₓₓ as well.

det H = fₓₓ fᵧᵧ − fₓᵧ²

Worked examples

For f(x,y) = 11x² + (-7)xy + 11y², calculate the determinant of its Hessian. Round the final result to 4 decimal places if needed.

H = [[22,-7],[-7,22]]. Take the product of its diagonal entries minus the product of its off-diagonal entries.

435

For f(x,y) = 7x² + (7)xy + 10y², calculate the determinant of its Hessian. Round the final result to 4 decimal places if needed.

H = [[14,7],[7,20]]. Take the product of its diagonal entries minus the product of its off-diagonal entries.

231

Try it

  1. For f(x,y) = 11x² + (0)xy + 1y², calculate the determinant of its Hessian. Round the final result to 4 decimal places if needed.

  2. For f(x,y) = 11x² + (3)xy + 10y², calculate the determinant of its Hessian. Round the final result to 4 decimal places if needed.

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Answers and marking notes
  1. 44

    H = [[22,0],[0,2]]. Take the product of its diagonal entries minus the product of its off-diagonal entries.

  2. 431

    H = [[22,3],[3,20]]. Take the product of its diagonal entries minus the product of its off-diagonal entries.

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