Suppose f: R → R is continuously differentiable. Show that if f'(x) > 0 for xo R, then there exists some interval I = (xo - 8, xo + 6) such that f|, : I → j bijective.
Suppose f: R → R is continuously differentiable. Show that if f'(x) > 0 for xo R, then there exists some interval I = (xo - 8, xo + 6) such that f|, : I → j bijective.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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This problem introduces a very reduced version of the inverse function theorem.
![Suppose \( f : \mathbb{R} \to \mathbb{R} \) is continuously differentiable. Show that if \( f'(x_0) > 0 \) for some \( x_0 \in \mathbb{R} \), then there exists some interval \( I = (x_0 - \delta, x_0 + \delta) \) such that \( f|_I : I \to f(I) \) is bijective.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F97e612ef-1556-436b-b62c-352b280e9e69%2F4515431d-8356-4eaf-aa13-36d1439a7d70%2Fo5rkyd_processed.png&w=3840&q=75)
Transcribed Image Text:Suppose \( f : \mathbb{R} \to \mathbb{R} \) is continuously differentiable. Show that if \( f'(x_0) > 0 \) for some \( x_0 \in \mathbb{R} \), then there exists some interval \( I = (x_0 - \delta, x_0 + \delta) \) such that \( f|_I : I \to f(I) \) is bijective.
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