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This problem illustrates that the derivative of a differentiable function need not be differentiable. Let Ja if x + 0 and |0, if x = 0. f(x) = (a) For any x #0, show that f(x) is differentiable at x and find a formula (with justification of course) for f'(x). (b) Show f is differentiable at 0 and find f' (0). (c) Parts (a) and (b) show that f'(x) is defined on all of R. Prove f'(x) is not differentiable at 0.

This problem illustrates that the derivative of a differentiable function need not be differentiable. Let Ja if x + 0 and |0, if x = 0. f(x) = (a) For any x #0, show that f(x) is differentiable at x and find a formula (with justification of course) for f'(x). (b) Show f is differentiable at 0 and find f' (0). (c) Parts (a) and (b) show that f'(x) is defined on all of R. Prove f'(x) is not differentiable at 0.

Advanced Engineering Mathematics
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 illustrates that the derivative of a differentiable function need not be differentiable. Let if x +0 and f(x) = |¤| if x = 0. (a) For any x 70, show that f (x) is differentiable at x and find a formula (with justification of course) for f'(x). (b) Show f is differentiable at 0 and find f' (0). (c) Parts (a) and (b) show that f'(x) is defined on all of R. Prove f'(x) is not differentiable at 0.
Transcribed Image Text:This problem illustrates that the derivative of a differentiable function need not be differentiable. Let if x +0 and f(x) = |¤| if x = 0. (a) For any x 70, show that f (x) is differentiable at x and find a formula (with justification of course) for f'(x). (b) Show f is differentiable at 0 and find f' (0). (c) Parts (a) and (b) show that f'(x) is defined on all of R. Prove f'(x) is not differentiable at 0.
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