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Suppose f is differentiable at a and f(a) + 0. (a) Prove |f| is differentiable at a by using that |f| = /f². What are the possible values of |f|'(a)? (b) Give a counterexample that shows the conclusion need not hold if f(a) = 0. Identify where your calculations for (a) fail in this case.

Suppose f is differentiable at a and f(a) + 0. (a) Prove |f| is differentiable at a by using that |f| = /f². What are the possible values of |f|'(a)? (b) Give a counterexample that shows the conclusion need not hold if f(a) = 0. Identify where your calculations for (a) fail in this case.

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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Suppose f is differentiable at a and f(a) # 0. (a) Prove |f| is differentiable at a by using that |f| = Vf2. What are the possible values of |fl'(a)? (b) Give a counterexample that shows the conclusion need not hold if f(a) = 0. Identify where your calculations for (a) fail in this case.
Transcribed Image Text:Suppose f is differentiable at a and f(a) # 0. (a) Prove |f| is differentiable at a by using that |f| = Vf2. What are the possible values of |fl'(a)? (b) Give a counterexample that shows the conclusion need not hold if f(a) = 0. Identify where your calculations for (a) fail in this case.
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I don't understand this step. I think if should be |f(x)-f(a)|<e when |x-a|<d.

f(x) - f(a) x-a whenever 1x-a) < S - M) < ein <E,
Transcribed Image Text:f(x) - f(a) x-a whenever 1x-a) < S - M) < ein <E,
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