Question

求详解！！！，书上的答案太简陋了，张宇一千题里的

Answer 1

f(x) cts, f(0)=0
F(x) = ∫(0->x) t^(n-1) . f( x^n-t^n) dt
To find : lim(x->0) F(x)/x^(2n)
Solution :
let
u= x^n - t^n
du = -nt^(n-1) dt
t=0, u= x^n
t=x, u= 0
∫(0->x) t^(n-1) . f( x^n-t^n) dt
=-(1/n)∫(x^n->0) f(u) du
= (1/n)∫(0->x^n) f(u) du
lim(x->0) F(x)/x^(2n)
=lim(x->0) (1/n)∫(0->x^n) f(u) du /x^(2n) ;(0/0 分子分母分别求导）
=lim(x->0) (1/n)[ nx^(n-1) ]f(x^n) / [(2n).x^(2n-1)]
=lim(x->0) f(x^n) / [(2n).x^n]
=[1/(2n) ] lim(x->0) f(x^n) / x^n
=[1/(2n) ] f'(x^n)