Consider the function defined by f(x)= -2x³-3x²+5x-4.\r
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document.write( "Use the intermediate value theorum to show that f has a zero between 0 and -1.
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document.write( "Sorry but f does not have a zero between 0 and -1.\r\n" );
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document.write( "Its graph looks like this and as you can see it doesn't cross \r\n" );
document.write( "the x-axis between 0 and -1, so it does not have a zero there.\r\n" );
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document.write( "You should point that out to your teacher. It must have been a typo.\r\n" );
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document.write( "However f does have a zero between -2 and -3 because \r\n" );
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document.write( "INTERMEDIATE VALUE THEOREM\r\n" );
document.write( "Let a and b be real numbers such that a < b. If f is a polynomial function such\r\n" );
document.write( "that f(a) and f(b) are opposite in sign, then there exists at least one zero in\r\n" );
document.write( "the interval (a, b).\r\n" );
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document.write( "[Notice that the other tutor, although he was lucky to have been correct in\r\n" );
document.write( "saying f has no zero between 0 and -1, he did so by incorrectly assuming the\r\n" );
document.write( "converse of the intermediate value theorem, which is not necessarily true.] \r\n" );
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document.write( "Here a = -3 and b = -2. f(-3) = 8 which is positive and f(-2) = -10, \r\n" );
document.write( "which is negative. They are opposite in sign. Therefore, by the above theorem,\r\n" );
document.write( "there exists at least one zero in the interval [a, b] which is the interval\r\n" );
document.write( "(-3,-2).\r\n" );
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document.write( "The reasoning is that since the graph of a polynomial has no \"loose ends\", if\r\n" );
document.write( "it is above the x axis at one place and below it at another place, it must\r\n" );
document.write( "have crossed over the x-axis at least once between them. It's like if you know\r\n" );
document.write( "that somebody was in New York at noon and in New Jersey at 1 PM on the same\r\n" );
document.write( "day, then they must have crossed the NY-NJ state line sometime between noon and\r\n" );
document.write( "1 PM (since 1 hour is not enough time for them to have gone all the way around\r\n" );
document.write( "to NJ through another state.) The other tutor's incorrect reasoning amounted to\r\n" );
document.write( "\"If a person is in NY at 12 noon and also in NY at 1 PM on the same day, then\r\n" );
document.write( "they could not have crossed the NY-NJ state line during that hour\", which is\r\n" );
document.write( "false reasoning because they could have gone to NJ, turned around and come back\r\n" );
document.write( "to NY, crossing the state line twice.)\r\n" );
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document.write( "Edwin
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