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Rog_Sec_14_7_P_1

Rog_Sec_14_7_P_1 - Math 20C Lecture Examples Section 14.7...

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(8/18/08) Math 20C. Lecture Examples. Section 14.7, Part 1. Maxima and minima: The first-derivative test Definition 1 (a) A function f ( P ) of two or three variables has a global maximum at a point P 0 if f ( P ) f ( P 0 ) for all points P in its domain. It has a global minimum at P 0 if f ( P ) f ( P 0 ) for all P in its domain. (b) The function has a local maximum at P 0 if it is defined in an open circle (in the two-variable case) or a open sphere (in the three-variable case) centered at P 0 and f ( P ) f ( P 0 ) for all points P in the circle or sphere . The function has a local minimum at P 0 if f ( P ) f ( P 0 ) for all points P in such a circle or sphere. Recall that x = a is a critical point of a function y = f ( x ) of one variable if the function is defined in an open interval containing a and if either f prime ( a ) is zero or f prime ( a ) does not exist. The definition of critical points for functions of two or three variables is similar. Definition 2 (a) The point P ( a, b ) is a critical point of z = f ( x, y ) if the function is defined in an
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