Prove that |sinh y - sinh x| ≤ (cosh a)|y - x|

Jan 2010
50
0
Let a > 0.
Let x ϵ (0,a) and y ϵ (0,a).
Use the fact that cosh is a primitive of sinh to prove that
|sinh y - sinh x| ≤ (cosh a)|y - x|

Thanks very much for any help.
 
Last edited:

Plato

MHF Helper
Aug 2006
22,458
8,632
Use the mean value on \(\displaystyle \sinh(t)\) on the interval \(\displaystyle [x,y]\).
Note that \(\displaystyle \cosh(c)\le \cosh(a)\) if \(\displaystyle c<a\).