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I'm so bad with the Mean Value Theorem. Can someone help me prove that, if f(x)=sinx, that, for any given a and b, |sinb-sina|<=|b-a|. Explain if you could too, please. Thanks a lot.
x = y → |x| = |y|
|sin(b)-sin(a)| = |f'(c)(b-a)|
f'(x) = cosx so f'(c) = cos(c).
|sin(b)-sin(a)| = |cos(c)||b-a|. And -1<=cosx<=1 so would it be |sin(b)-sin(a)| <= 1|b-a| ?