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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: [tex]\left|[/tex]sin a -sin b [tex]\right|[/tex] [tex]\leq[/tex] [tex]\left|[/tex] a - b[tex]\right|[/tex] 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.
  2. jcsd
  3. Oct 28, 2008 #2
    The mean value theorem states that in an interval [a,b]:

    [tex]f'(c) = \frac{ \sin b - \sin a}{b-a} = \cos(c) [/tex]

    Now put absolute value signs there and make use of [tex]|\cos(x)| \leq 1[/tex]
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