Engineering Calculus Notes 377

Engineering Calculus Notes 377 - 365 3.8. LOCAL EXTREMA →...

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Unformatted text preview: 365 3.8. LOCAL EXTREMA → → →− • g has a strict minimum at the origin: g(− ) > 0 for all − = 0 , x x and • h has saddle behavior: its restriction to the x-axis has a minimum at the origin, while its restriction to the line y = −x has a maximum at the origin. As an example, consider the function f (x, y ) = 5x2 + 6xy + 5y 2 − 8x − 8y. We calculate the first partials fx (x, y ) = 10x + 6y − 8 fy (x, y ) = 6x + 10y − 8 and set both equal to zero to find the critical points: 10x + 6y = 8 6x + 10y = 8 has the unique solution (x, y ) = 11 , 22 . Now we calculate the second partials fxx (x, y ) = 10 fxy (x, y ) = 6 fyy (x, y ) = 10. Thus, the discriminant is ∆2 (x, y ) := fxx fyy − (fxy )2 = (10) · (10) − (6)2 > 0 and since also ∆1 (x, y ) = fx (x, y ) = 6 > 0 ...
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

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