Question

一道高数题追加50分求解

Answer 1

f(x)
=x^3.sin(1/x) ; x≠0
=0 ; x=0
lim(x->0) x^3.sin(1/x) =0 =f(0)
x=0, f(x) 连续
f'(0)
=lim(h->0) [h^3.sin(1/h) -f(0)]/h
=lim(h->0) h^2.sin(1/h)
=0
x≠0
f'(x)
=3x^2.sin(1/x) + x^3.cos(1/x) .(-1/x^2)
=3x^2.sin(1/x) - x.cos(1/x)
f''(0)
=lim(h->0) [3h^2.sin(1/h) - h.cos(1/h) -f'(0)]/h
=lim(h->0) [3h.sin(1/h) - cos(1/h)]
不存在
x=0 , 不是极值点