LimitsHard
Question
If f(x) is differentiable everywhere, then:
Options
A.|f| is differentiable everywhere
B. |f|2is differentiable everywhere
C. f | f | is not differentiable at some point
D. f + | f | is differentiable everywhere
Solution
If ′f′ is differentiable
then | f | is differentiable at each point x, where f(x) ≠ 0
if f(α) = 0 and f′(a) = 0, then | f | is differentiable at x = α
if f(α) = 0 and f′(a) ≠ 0, then | f | is not differentiable at x = α
⇒ If f is differentiable then | f | may or may not be
differentiable, [option A, C, D not necessarly true]
Now | f |2 = f2
(f 2)′ = 2.f.f′ since f is differentiable
∴ f2 is also differentiable
then | f | is differentiable at each point x, where f(x) ≠ 0
if f(α) = 0 and f′(a) = 0, then | f | is differentiable at x = α
if f(α) = 0 and f′(a) ≠ 0, then | f | is not differentiable at x = α
⇒ If f is differentiable then | f | may or may not be
differentiable, [option A, C, D not necessarly true]
Now | f |2 = f2
(f 2)′ = 2.f.f′ since f is differentiable
∴ f2 is also differentiable
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