Question

不是f(0)=0嘛 为什么原函数也等于0？

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Answer 1

lim(x-0) f(x)/x =1

g(x) =∫(0->1) f(xt) dt

To find: g'(x)

To prove: x=0, g'(x)  连续

solution:

f(0) =0

g(x) =∫(0->1) f(xt) dt

g(0) 

=∫(0->1) f(0) dt

=∫(0->1) 0 dt

=0

//

let

u=xt

du= xdt

t=0, u=0

t=1, u=x

g(x) 

=∫(0->1) f(xt) dt

=∫(0->x) f(u) (du/x)

=(1/x)∫(0->x) f(u) du

=(1/x)∫(0->x) f(t) dt

x≠0

g'(x) =(1/x)f(x) -(1/x^2)∫(0->x) f(t) dt

lim(x->0) g'(x)

=lim(x->0)  [(1/x)f(x) -(1/x^2)∫(0->x) f(t) dt ]

=lim(x->0)  [xf(x) -∫(0->x) f(t) dt ]/ x^2

洛必达

=lim(x->0)  [xf'(x)+f(x) -f(x) ]/ (2x)

=lim(x->0)  xf'(x)/ (2x)

=(1/2)f'(0)

x=0

g'(0)

=lim(h->0) [g(h) -g(0) ]/h

=lim(h->0) [(1/h)∫(0->h) f(t) dt -g(0) ]/h

=lim(h->0) ∫(0->h) f(t) dt /h^2

洛必达

=lim(h->0)  f(h)  /(2h)

洛必达

=lim(h->0)  f'(h)/2

=(1/2)f'(0)

=lim(x->0) g'(x)

x=0, g'(x) 连续

Answer 2

请看清题目，g(x)的表达式里把x=0代入，f(xt)就是f(0)等于0

Answer 3

g(0)=∫(0到1)f(0)dt
=∫(0到1) 0 dt
=0 ∫(0到1）dt
=0