Please see the attached file for the fully formatted problem.

Let > 0. Prove that log x x for x large. Prove that there exists a
constant C such that log x Cx for all x 2 [1, 1 ), C ! 1 as ! 0+,
and C ! 0 as ! 1 Please justify all steps and be rigorous because it is an analysis problem.
(Note: The problem falls under the chapter on Differentiability on R in
the section entitled The Mean Value Theorem, and the hint says: Find the
maximum of f(x) = log x/x forx 2 [1, 1 ))
1

For numbers a1,....,an, define
p(x) = a1x +a2x^2+....+anx^n for all x. Suppose that:
(a1)/2 + (a2)/3 +....+ (an)/(n+1) = 0
Prove that there is some point x in the interval (0,1) such that p(x) = 0

Context: We are learning Rolle, Lagrange, Fermat, Taylor Theorems in our Real Analysis class. We just finished continuity and are now studying differentiation.
Question:
Let f: [a,b] --> R, a < b, twice differentiable with the second derivative continuous such that f(a)=f(b)=0.
Denote M = sup |f "(x)| where x is in [a,b]

Show that it is impossible to write R=U(union sign n=1 bottom, infinity top)F_n where for each n belong to N, F_n is closed set containing no nonempty open intervals.

1) Show that the real part of the function z^(1/2) is always positive.
2) Suppose f: G --> C ( C complex plane) is analytic and that G is connected. Show that if f(z) is real for all z in G, then f is a constant.

Let f = u + iv be an analytic function on an open connected set G in C ( C = complex plane) where u and v are its real and imaginary parts. assume u(z) >= u(a) for some a in G and all z in G. Prove that f is constant.

We have learned Rolle, Lagrange, Fermat, Taylor Theorems in our RealAnalysis class and we have finished differentiation. We just started integration. In this problem we are not supposed to use any material we haven't learned, ie integration. We are using the books by Rudin, Ross, Morrey/Protter.
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Expand the given sketch of a proof into a reasonably complete proof.
It is suggested that the following proof be used:
Maximum Modulus Principle: Let U be a non-empty open subset of the complex numbers C, let D(z0,r)={z є C: |z-z0| ≤ r} be a closed disk (with center z0 and radius r >0) that is entirely contained in U, le

For any two real numbers a and b, a < b if and only there exist a positive real number s such that a + s = b. Use this definition to prove that for any negative real number r, if a < b then a + r < b + r.