(MVT) f(x)=sinx. Show that, for any given a and b, |sina-sinb|<=|b-a|

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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.
 

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  • #2
gb7nash
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Start with what you know. What does the mean value theorem say?
 
  • #3
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First of all, I just realized I should've posted this in the homework section. And I know the function is cont. on [a,b] and f'(x) exists on (a,b) so f'(c)=(f(b)-f(a))/(b-a). So f'(c)=(sinb-sina)/(b-a). But I'm not sure how f'(c) comes in to play. And I don't know how to interperate the absolute value either.
 
  • #4
gb7nash
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Ok, so we have:

(sin(b)-sin(a)) = f'(c)(b-a)

It's a well-known fact that x = y → |x| = |y|.

How can we apply that?
 
  • #5
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I honestly don't know...lol :\
 
  • #6
gb7nash
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x = y → |x| = |y|
This says that:

left-hand side = right-hand side

can be rewritten as:

|left-hand side| = |right-hand side|

For example:

a+b = c+d → |a+b| = |c+d|

Now, let's look at the problem at hand. We have:

(sin(b)-sin(a)) = f'(c)(b-a)

Applying the absolute value on both sides, how can we rewrite this?
 
  • #7
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|sin(b)-sin(a)| = |f'(c)(b-a)|. I understand this, I just read it wrong, lol sorry
 
  • #8
gb7nash
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|sin(b)-sin(a)| = |f'(c)(b-a)|
Ok, good. Now, let's calculate what f'(c) is. What's f'(x)? After you have that, plug c in.
 
  • #9
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f'(x) = cosx so f'(c) = cos(c).
 
  • #10
gb7nash
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f'(x) = cosx so f'(c) = cos(c).
Correct. So now, we have:

|sin(b)-sin(a)| = |cos(c)(b-a)|

It's also a well-known fact that |xy| = |x||y|. How could we rewrite the right-hand side?
 
  • #11
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|sin(b)-sin(a)| = |cos(c)||b-a|. And -1<=cosx<=1 so would it be |sin(b)-sin(a)| <= 1|b-a| ?
 
  • #12
gb7nash
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|sin(b)-sin(a)| = |cos(c)||b-a|. And -1<=cosx<=1 so would it be |sin(b)-sin(a)| <= 1|b-a| ?
Correct.
 
  • #13
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Thank you so much for working with me and not just posting an unexplained proof. Props!
 

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