Question

高等数学题，求大神解答

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Answer 1

f(x) = x^n.sin(1/x) ,x≠0
=0 , x=0

(1)
=lim(x->0) f(x)
=lim(x->0) x^n.sin(1/x) =f(0) =0
=> n≥1
(2)
f(0) 连续
=> n≥1 (1)
f'(0)
=lim(h->0)[ h^n.sin(1/x) - f(0)]/h
=lim(h->0) h^(n-1).sin(1/x)
=>
n-1 ≥1
n≥2 (2)
(1) and (2)
=>n≥2
(3)

f'(x) = nx^(n-1).sin(1/x) - x^(n-2). cos(1/x) ,x≠0
= 0 , x=0

lim(x->0) f'(x)
=lim(x->0) [ nx^(n-1).sin(1/x) - x^(n-2). cos(1/x) ]
=>
n-1≥1 and n-2≥1
=>n≥3

f'(0) 连续 ： n≥3