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(e) Using parts (a) and (b), show that the equation tan(3r) = √2r+1 has a unique solution on (0, 1). Hint: Formulate this equation as z=f(z) for some function f(x). Then use the results from part (a) and part (b).

(e) Using parts (a) and (b), show that the equation tan(3r) = √2r+1 has a unique solution on (0, 1). Hint: Formulate this equation as z=f(z) for some function f(x). Then use the results from part (a) and part (b).

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter4: Polynomial And Rational Functions
Section4.6: Variation
Problem 15E
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Please answer Part C of this question. Thank you.

3. In this problem we study another approach to show that certain equations have a unique solution on an interval [a, b]. The goal is to first write the equation in the form x = f(x) for some function f defined on [a, b]. (a) Suppose that f is continuous on [a, b] and that a < f(x) < b for all x € [a, b]. Prove that there exists c = (a, b) such that c= f(c). Hint: Consider the function g(x) = x-f(x). (b) Suppose, in addition, that f is differentiable on (a, b) and that f'(x)| < 1 for all x € (a, b). Prove that there is a unique point c € (a, b) such that c= f(c). Hint: Use MVT. (c) Using parts (a) and (b), show that the equation tan(3r) = √2x+1 has a unique solution on (0, 1). Hint: Formulate this equation as x = f(x) for some function f(x). Then use the results from part (a) and part (b).
Transcribed Image Text:3. In this problem we study another approach to show that certain equations have a unique solution on an interval [a, b]. The goal is to first write the equation in the form x = f(x) for some function f defined on [a, b]. (a) Suppose that f is continuous on [a, b] and that a < f(x) < b for all x € [a, b]. Prove that there exists c = (a, b) such that c= f(c). Hint: Consider the function g(x) = x-f(x). (b) Suppose, in addition, that f is differentiable on (a, b) and that f'(x)| < 1 for all x € (a, b). Prove that there is a unique point c € (a, b) such that c= f(c). Hint: Use MVT. (c) Using parts (a) and (b), show that the equation tan(3r) = √2x+1 has a unique solution on (0, 1). Hint: Formulate this equation as x = f(x) for some function f(x). Then use the results from part (a) and part (b).
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