Multivariable extrema
Objective. Classify critical points using the Hessian.
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- 2. Worked example
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Learn: the key idea
Multivariable extrema are the local maxima, minima and saddle points of a function of several variables, found first by locating critical points where ∇f = 0. To classify each critical point, the Hessian matrix of second partial derivatives is used: compute D = fₓₓfᵧᵧ − (fₓᵧ)² at the point; if D > 0 and fₓₓ > 0 it is a local minimum, if D > 0 and fₓₓ < 0 it is a local maximum, and if D < 0 the point is a saddle. When D = 0 the test is inconclusive and other methods are needed. For f(x, y) = x² + y², the only critical point (0, 0) gives D = (2)(2) − 0² = 4 > 0 with fₓₓ = 2 > 0, confirming a local minimum.
Worked example
Classify the critical point (0, 0) of f(x, y) = x² − y² using the Hessian test for multivariable extrema.
- 1Find the second partial derivatives: fₓₓ = 2, fᵧᵧ = −2, fₓᵧ = 0.
- 2Compute the discriminant D = fₓₓfᵧᵧ − (fₓᵧ)² = (2)(−2) − 0².
- 3Simplify: D = −4.
- 4Since D < 0, classify the critical point using the saddle point rule.
Answer: (0, 0) is a saddle point
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Practice
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Start practisingCommon mistake. Students classify a critical point using only the sign of fₓₓ without computing the full discriminant D = fₓₓfᵧᵧ − (fₓᵧ)²; ignoring the mixed partial term can wrongly label a saddle point as an extremum.
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