Math_137_Winter_2010_Week_8_Notes

Math_137_Winter_2010_Week_8_Notes - Math 137 Week 8 Notes...

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Math 137 Week 8 Notes – Dr. Paula Smith 4.1 We often want to optimize a function, that is, find its greatest or smallest value. A function has an absolute (or global) maximum at c if f(c) f(x) for all x in Dom f. The number f(c) is the maximum value of f on D; c is a maximum point . There can be only one maximum value, but there can be many maximum points. Similarly, a function has an absolute (or global) minimum at c if f(c) f(x) for all x in Dom f. The number f(c) is the minimum value of f on D; c is a minimum point . There can be only one minimum value, but there can be many minimum points. A minimum or maximum is an extremum . The plural of each word is minima, maxima, extrema. A function f has a local (or relative) maximum at c if if f(c) f(x) when x is near c (for all x in some open interval containing c). Similarly, f has a local (or relative) minimum at c if if f(c) f(x) when x is near c. An absolute extremum is also a local extremum, but a local extremum is not necessarily an absolute extremum. The Extreme Value Theorem: If f is continuous on a closed interval [a,b] then f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some numbers c and d in [a,b]. The point of this theorem is that we are guaranteed that extrema exist as long as f is continuous and we are investigating a closed interval. However, it doesn’t tell us where they are. Fermat’s Theorem tells us that if f has a local extremum at point c, then f (c) either is 0 or doesn’t exist. However, not every point c such that f (c) is 0 or f
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This note was uploaded on 04/13/2010 for the course MATH 137 taught by Professor Speziale during the Spring '08 term at Waterloo.

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Math_137_Winter_2010_Week_8_Notes - Math 137 Week 8 Notes...

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