Yahoo Answers: Answers and Comments for Integration Proof Question? [Mathematics]
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From Duncan
enGB
Mon, 25 Feb 2019 20:42:13 +0000
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Yahoo Answers: Answers and Comments for Integration Proof Question? [Mathematics]
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https://uk.answers.yahoo.com/question/index?qid=20190225204213AAbrWkO
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From az_lender: If f(x) < x for ALL x in [0,1], then the ar...
https://uk.answers.yahoo.com/question/index?qid=20190225204213AAbrWkO
https://uk.answers.yahoo.com/question/index?qid=20190225204213AAbrWkO
Mon, 25 Feb 2019 22:03:42 +0000
If f(x) < x for ALL x in [0,1], then the area would be less than 1/2, because the graph of f would lie entirely below the top edge of the triangle described by f(x) = x on [0,1].
If f(x) > x for ALL x in [0,1], then the area would be greater than 1/2, same reasoning.
If f(x) = x for all x in [0,1], there is nothing to prove, we are finished.
The only remaining cases are cases where SOME x's in [0,1] have f(x) less than x, while some other x's in [0,1] have f(x) greater than x. But then the graph of (a continuous) f(x) must cross the line f(x) = x, somewhere.