Question

高数，第十三题怎么做？

Answer 1

(13)
f(x)
=x^2.sin(1/x^2) ; x≠0
=0 ; x=0
lim(x->0) x^2.sin(1/x^2) =0 =f(0)
x=0, f(x) 连续
f'(0)
=lim(h->0) [h^2.sin(1/h^2) -f(0)]/h
=lim(h->0) h.sin(1/h^2)
=0
x∈ [-1,0 ) U (0, 1]
f(x) =x^2.sin(1/x^2)
f'(x)
=x^2.cos(1/x^2) .(-2/x^3) + 2x.sin(1/x^2)
=-(2/x)cos(1/x^2) + 2x.sin(1/x^2)
lim(x->0+) [-(2/x)cos(1/x^2) + 2x.sin(1/x^2)] ->-∞
lim(x->0-) [-(2/x)cos(1/x^2) + 2x.sin(1/x^2)] ->+∞
=>

f(x) 在 [-1，1] 上处处可导, 但f'(x) 在 [-1，1] 上无界