Edition of Understanding Analysis Assignment | Assignment Help Services

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

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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.

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Exercise 5.3.4

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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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