设f‘(x)在[a,b]上连续,且f(a)=0,证明:|∫b a f(x)dx|求详细过程

来源：学生作业帮助网编辑：六六作业网时间：2024/12/18 14:41:27

设f‘(x)在[a,b]上连续,且f(a)=0,证明:|∫baf(x)dx|求详细过程设f‘(x)在[a,b]上连续,且f(a)=0,证明:|∫baf(x)dx|求详细过程设f‘(x)在[a,b]上连

设f‘(x)在[a,b]上连续,且f(a)=0,证明:|∫b a f(x)dx|求详细过程
设f‘(x)在[a,b]上连续,且f(a)=0,证明:|∫b a f(x)dx|
求详细过程

设f‘(x)在[a,b]上连续,且f(a)=0,证明:|∫b a f(x)dx|求详细过程
设g(x) = ∫ f(t)dt,则g'(x) = f(x),g"(x) = f'(x).
g(x)在[a,b]二阶连续可导,且g(a) = 0,g'(a) = f(a) = 0.
由带Lagrange余项的Taylor展开,存在c∈(a,b)使
g(b) = g(a)+g'(a)(b-a)+g"(c)(b-a)²/2 = f'(c)(b-a)²/2.
即有| ∫ f(t)dt| = |g(b)| = |f'(c)|·(b-a)²/2 ≤ max{|f'(x)|}·(b-a)²/2.

设f(x)在[a,b]上连续,且a 设f(x)在[a,b]上连续,且a 设f(x)在[a,b]上连续,且a 设函数f(x)在[a,b]上连续,在(a,b)内可导且f'(x) 设函数f(x)在[a,b]上连续,在(a,b)上可导且f'(x) 设函数f(x),g(x)在区间[a,b]上连续,且f(a) 证明设f(x)在有限开区间(a,b)内连续,且f(a+) ,f(b-)存在,则f(x)在(a,b)上一致连续. 设函数f(x)在[a,b]上连续,在(a,b)可导,且f(a)*f(b)>0,f(a)*f((a+b)/2) 设f(x) 在[a,b] 上连续,且f(x)>0.求证:∫(a,b)f(x)dx*∫(a,bdx/f(x)≥(b-a)^2. 设函数f(x)在[a,b]上连续,a 设f(x)在[a,b]上连续,a 设函数f(x)在[a,b]上连续,a 设f(x)在[a,b]上连续,且没有零点,证明f(x)在[a,b]上保号设f(x)在闭区间（a,b)上连续,且a 设函数f(x)在[a,b]上连续,且a 设函数f(x)在[a,b]上连续,且a 设函数f(x)在[a,b]上连续,在(a,b)内有二阶导数,且有f(a)=f(b)=0,f(c)>0(a 设f'(x)在[a,b]上连续,在（a,b)内二阶可导,且f(a)=f(b)=0,f(c)>0,a