Initialize
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Tools≻Load Package:
Student Multivariate Calculus
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Loading Student:-MultivariateCalculus
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Context Panel: Assign Name
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Find critical points via first principles
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Calculus palette: Partial derivative operator
Press the enter key.
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Context Panel: Solve≻Solve
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Alternate calculation of the critical points
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Type and press the Enter key.
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Context Panel: Student Multivariate Calculus≻
Differentiate≻Gradient
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Context Panel: Conversions≻To List
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Context Panel: Conversions≻Equate to 0
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Context Panel: Solve≻Solve
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Obtain
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Expression palette: Evaluation template
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Context Panel: Evaluate and Display Inline
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=
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To apply the Sylvester criterion, the Hessian must be obtained. Unfortunately, there is no syntax-free way to obtain this matrix.
Obtain the Hessian and determine that it is positive definite
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Apply the Hessian command from the VectorCalculus package.
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Context Panel: Queries≻Is Definite?≻Positive Definite?
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=
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Since the Hessian is a positive definite matrix, the second-derivative term in the Taylor series expansion of simply adds to 2, the value of at the critical point. So, near the critical point, function values are greater than 2, making the critical point a local minimum. Sylvester's criterion will determine that is positive definite, and hence that the critical point is a minimum.
The sequence = is easily obtained for , which is a diagonal matrix. Since there are no sign changes in the sequence, the critical point is a local (relative) minimum by the terms of Sylvester's criterion.