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3. Consider the function f: R→R, f(x) = = S x + 2x² sin(1/x²) for x ‡0 | 0 if x = 0 Prove that the derivative f'(0) is invertible, but f is not invertible in any interval (-ɛ, ɛ) for ɛ > 0.

3. Consider the function f: R→R, f(x) = = S x + 2x² sin(1/x²) for x ‡0 | 0 if x = 0 Prove that the derivative f'(0) is invertible, but f is not invertible in any interval (-ɛ, ɛ) for ɛ > 0.

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter5: Graphs And The Derivative
Section5.1: Increasing And Decreasing Functions
Problem 44E: Where is the function defined by f(x)=ex increasing? Decreasing? Where is the tangent line...
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Please write a full-detailed solution for this Analysis II problem, I would appreciate it since I need to understand it cause I am confused and stuck on this one. Will give like!

3. Consider the function f: R → R, f(x) = x + 2x² sin(1/x²) for x ‡ 0 0 if x = 0 Prove that the derivative f'(0) is invertible, but f is not invertible in any interval (-e, e) for ε > 0.
Transcribed Image Text:3. Consider the function f: R → R, f(x) = x + 2x² sin(1/x²) for x ‡ 0 0 if x = 0 Prove that the derivative f'(0) is invertible, but f is not invertible in any interval (-e, e) for ε > 0.
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