By Phil Dyke (auth.)

ISBN-10: 0333660927

ISBN-13: 9780333660928

ISBN-10: 1349140767

ISBN-13: 9781349140763

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Finally, if /yy = 0 and the other two second-order partial derivatives of f(x, y) are non-zero, then once again there is a saddle point unless fxy = 0, in which case there is a maximum if fxx < 0 and a minimum if fxx > 0. Y = 38 These must simultaneously be zero. We therefore solve the equations 2x-2xy=O 4y- r = o The first equation gives x = 0 or y = 1. If x = 0, the second equation gives y = 0, whence (0, 0) is an extreme (or stationary) value. If y = 1 then the second equation gives x 2 = 4 so x = ± 2 giving two further stationary values (2, 1) and ( -2, 1).

0 , i = 1, 2, ... , n Thus = 52 1 -- 2A. VJ which shows that the vector h must be parallel to Vf at the location x = x 0 • It will be shown (in Chapter 7) that, in three dimensions, the vector Vf is always perpendicular to the contours of f. That Vf points to the direction of steepest descent is therefore no surprise. 2 Solution Use matrix notation to determine Taylor's Theorem for a function of n variables In two variables, Taylor's Theorem is j(a + h, b + k) = j(a, b) + = + (h 1 ( h CJx () n!

Hence, the area S of the trolley = pq = q{(b - qcos9)cosec9 + (a - qsin9)sec9}. In order to achieve a minimum or maximum, we impose the conditions on S(q,9): as aq = 0 ::::} bcosece - 2qcos9 cosece + asece - 2qsin9 sece = 0, and as = 0::::} -bcosec9cot9 + qcosec 2 9 + asec9tan 9- qsec 2 9 = 0. We solve these two equations ae for 9 and q. Some trigonometric manipulation is required before we arrive at the following simplification: bcos9 + asin9 2q q cos29 and Substituting for q into the expression for p((b - qcos9)cosec9 + (a - qsin9)sec9), gives on simplifying pcos9sin9 = q and q = t

### Advanced Calculus by Phil Dyke (auth.)

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