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The sign of 4ac − b² classifies a quadratic; the second derivative test carries that to any function. Name the second partial derivatives at a critical point A = f_xx, B = f_xy, C = f_yy, and the sign of AC − B² decides minimum, maximum or saddle. We see where the quadratic formula’s discriminant comes in, check the test on the quadratic, and justify it with the second-order Taylor approximation.
Key concepts covered:
– 4ac − b² as the discriminant of a quadratic in t = x/y
– Second partial derivatives f_xx, f_yy and the mixed partials f_xy = f_yx
– The test: AC − B² positive with A positive is a minimum, with A negative a maximum; AC − B² negative is a saddle
– Checking the test on w = ax² + bxy + cy²
– Why it works: the quadratic Taylor approximation of Δf at a critical point
– Degenerate points where AC − B² = 0 and the higher-order terms decide
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SOURCE MATERIALS
The source materials for this video are from https://www.youtube.com/watch?v=3_goGnJm5sA
Attribution: MIT OpenCourseWare, 18.02 Multivariable Calculus, Fall 2007, Prof. Denis Auroux — https://ocw.mit.edu/courses/18-02-multivariable-calculus-fall-2007/ — License: CC BY-NC-SA 4.0
