proof of Bernoulli’s inequality employing the mean value theorem
Observe that for all in fixed, is, indeed, differentiable on . In particular,
Consider two points in and in . Then clearly by the mean value theorem, for any arbitrary, fixed in , there exists a in such that,
Since is in , it is clear that if , then
and, accordingly, if then
Thus, in either case, from 1 we deduce that
if . From this we conclude that, in either case,. That is,
for all choices of in and all choices of in . If in , we have
for all choices of in . Generally, for all in and all in we have:
This completes the proof.
Notice that if is in then the inequality would be reversed. That is:
. This can be proved using exactly the same method, by fixing in the proof above in .
|Title||proof of Bernoulli’s inequality employing the mean value theorem|
|Date of creation||2013-03-22 15:49:53|
|Last modified on||2013-03-22 15:49:53|
|Last modified by||rspuzio (6075)|