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Smooth functions

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Let f be a function from N to R+ . Function f is eventually nondecreasing if
such that . Function f is smooth if it is eventually
nondecreasing and such that .
(a) Is the function + such that a smooth function? Verify!
(b) Is the function + such that = a smooth function? Verify!

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a.)
<br><br>because,
<br><br>f(n) = n*log(n), n belong to N.
<br><br>for m,n belong to N and m <= n:
<br><br>f(m) = m*lg(m)
<br><br>and,
<br><br>f(n) = n*lg(n)
<br><br>because, lg(n) is slow increasing function, therefore,
<br><br>f(n) >= f(m)
<br><br>=> f(n) is non-decreasing function Ok.
<br><br>Now,
<br><br>for k,n ...

Solution Summary

Smooth functions are noted.

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