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Prove |cosa - cosb| <= |a-b|

  • Thread starter FlorenceC
  • Start date
1. The problem statement, all variables and given/known data
I have no idea how to approach this question.

2. Relevant equations

3. The attempt at a solution
I suppose ∫ |cosa - cosb| < = |a-b|
sinb-sina <= b^2/2 - a^2/2
but now what do I do?
nvm. I figured it out, it's a subtle trick with MVT
Since it looks like you've found a solution ... assuming ##a\leq b## gives $$\left|\int_a^b\sin x\ \mathrm{d}x\right|\leq\int_a^b|\sin x|\ \mathrm{d}x\leq\int_a^b1\ \mathrm{d}x$$.
If ##a>b##, you just need to flip the limits on the last two integrals. The desired inequality isn't too incredibly difficult to get from there.

I like the MVT proof better, though. It's, like, one step.

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