Prove that f is differentiable, math homework help

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gbenycngry1995

Mathematics

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2. (10 points) Let a > 1,8 > 0, consider the function sin (2) f(3) = I >0; 1 0, 3+1. 2-0
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1

𝑓(𝑥) = {

𝑥 𝛼 sin (𝑥 𝛽) , 𝑥 ≥ 0,
0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒.

𝛼 > 1, 𝛽 > 0.

a. Prove that 𝑓 is differentiable at 0 and 𝑓 ′ (0) = 0.
Proof. First, prove that 𝑓 is continuous at 0. Indeed, |𝑓(𝑥)| ≤ 𝑥 𝛼 → 0, 𝑥 → 0,
therefore 𝑓(𝑥) → 0 = 𝑓(0), 𝑥 → 0 (because 𝛼 > 0).

Now consider (by definition of 𝑓 ′ (0)) the expression
1
𝛼
𝑓(0 + ℎ) − 𝑓(0) ℎ sin (ℎ𝛽 )
1
=
= ℎ𝛼−1 sin ( 𝛽 ).



1

The limit of this when ℎ → 0 is 0 the same way: |ℎ𝛼−1 sin (ℎ𝛽)| ≤ ℎ𝛼−1 → 0, ℎ → 0 (here we use that
𝛼 > 1, 𝛼 − 1 > 0).∎

b. Prove that lim 𝑓 ′ (𝑥) = 0 if and only if 𝛼 > 𝛽 + 1.
𝑥→0

For this, consider 𝑓 ′ (𝑥) for 𝑥 > 0:


𝑓 ′ (𝑥) = (𝑥 𝛼 sin(𝑥 −�...


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