一道大一的微积分题目

题目

一道大一的微积分题目
Consider the function f(x) = xx; x > 0.
(a) Find f0(x) and f00(x).
(b) Solve f0(a) = 0.Is there any b > 0 such that f00(b) =
(c) It is clear that f0(x) < 0 for x < a and f0(x) > 0 for x > a.Therefore f is decreasing on
(0; a) and increasing on (a;1) and so f is not 1􀀀1.Let g(x) = xx; x > a.Explain why g
is 1􀀀1.
(d) Find (g􀀀1)0(4).

答案

(1)f'(x)=2x.f"(x)=2
(2)f"(x)=2为常数,不存在f"（b）=0
(3)f(x) = xx=0,x=0.a=0