REAL ANALYSIS These exercises are taken from Abbott's second edition of understanding analysis. Providing well written detailed solutions will yield...

REAL ANALYSIS

These exercises are taken from Abbott's second edition of understanding analysis. Providing well written detailed solutions will yield a large tip.

Exercise 5.3.2.

Let f be differentiable on an interval A. If f'(x) is not equal to 0 on A, show that f is one-to-one on A. Provide an example to show that the converse statement need not be true.

Exercise 5.3.4

Let f be differentiable on an interval A containing zero, and assume (x_n) is a sequence in A with (x_n)→0 and x_n = 0.

(a) If f(x_n) = 0 for all n∈N, show f(0) = 0 and f'(0) = 0.

(b) Add the assumption that f is twice-differentiable at zero and show that f"(0) = 0 as well. 

Exercise 5.3.6. (a) Let g : [0, a]→R be differentiable, g(0) = 0, and |g'(x)| ≤M for all x∈[0, a]. Show |g(x)| ≤Mx for all x∈[0, a].

(b) Let h : [0, a]→R be twice differentiable, h'(0) = h(0) = 0 and |h"(x)| ≤ M for all x∈[0, a]. Show |h(x)| ≤(Mx^{2})/2 for all x∈[0, a].

(c) Conjecture and prove an analogous result for a function that is differentiable three times on [0, a]. 

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