(a) Let \( f \) be differentiable on \( (a, b) \). Prove that if \( f'(x) \neq 0 \) for each \( x \in (a, b) \), then \( f \) has at most one zero in \( (a, b) \).(b) Let \( f \) be twice differentiable on \( (a, b) \). Prove that if \( f''(x) \neq 0 \) for each \( x \in (a, b) \), then \( f \) has at most two zeros in \( (a, b) \).

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Answered: (a) Letƒ be differentiable on (a, b).…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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If a twice differentiable function f has exactly three zeros, at x=1, x=2 and x=3, then which of the following can we necessarily conclude? Select all that apply. O f' has at least two zeros in the interval (1,3) . O f' has exactly one zero in the interval (1,3). O f' is never zero on the interval (1, 3). f" has at least one zero in the interval (1,3). f" is never zero on the interval (1 , 3) .
7. Suppose that f = u + iv is analytic and not identically constant.a) Show that uv is the real part of an analytic function.b) Show that u^2 + v^2cannot be the real part of any analytic function.
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(a) Letƒ be differentiable on (a, b). Prove that if f'(x) #0 for each x e (a, b), then ƒ has at most one zero in (a, b). (b) Let ƒ be twice differentiable on (a, b). Prove that if f"(x)#0 for each x € (a, b), then ƒ has at most two zeros in (a, b).