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such that the equation in the unknown 
:
admits 
roots: 
s.t. 
.
be a function, two times derivable on 
for which there exist 
such that![\[ \frac { f(b)-f(a) }{b-a} \neq f'(c) ,\]](http://latex.artofproblemsolving.com/a/3/1/a31861ef909a6e17ed8b495b9b9b16d887ba573f.png)
for all 
.
.
, 
and 
.
be complex numbers with 
. Find the greatest possible value of the expression 
be a positive root of the equation 
. Prove that the following sequence converges: 
(0<x<2π)
with domain ![$[0,2025]$](http://latex.artofproblemsolving.com/d/1/7/d17ac41d54c4b1888fb8bb43cf11b372691694a0.png)
satisfying 
and 
is strictly increasing on Compute smallest real M such that all functions in this set ,
.
have local extrema at 
respectively. find 
. Here 
are constants .
be the region in the complex plane enclosed by curve 
for 
. Compute perimeter of .
and it is indeed correct), but I don't know how to prove it:
.
and 
Prove that
