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Prove Inequality using Mean Value Theorem

  1. Oct 28, 2008 #1
    1. The problem statement, all variables and given/known data
    Essentially, the question asks to use the mean value theorem(mvt) to prove the inequality: abs(sina - sinb) [tex]\leq[/tex] abs(a - b) for all a and b


    3. The attempt at a solution

    I do not have a graphing calculator nor can I use one for this problem, so I need to prove that the inequality basically by proof. What I did was to look at the mvt hypotheses: if the function is continous and differetiable on closed and open on interval a,b, respectively. However, the problem I am having is that I am getting thrown off by the absolute values and the fact that I've never used mvt on inequalities. I know the absolute value of the sin will look like a sequence of upside-down cups with vertical tangents between them. Hints most appreciated.
     
    Last edited: Oct 28, 2008
  2. jcsd
  3. Oct 28, 2008 #2
    Assume a>b, then sina-sinb=(a-b)cosc, for some b<c<a, which gives the inequality with no problems.
     
  4. Oct 28, 2008 #3
    Wait, are you considering that abs(...) = absolute value of the sum?
     
  5. Oct 28, 2008 #4

    Dick

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    The mean value theorem tells you (sin(a)-sin(b))/(a-b)=sin'(c)=cos(c) for some c between a and b, as boombaby said. Take the absolute value of both sides and use that |cos(c)|<=1.
     
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