设f（x)在区间[0,1]上连续，证明:

设函数fz)在[0,1]上连续,在(0,1)内可导,且<img src='https://img2.soutiyun.com/ask/2020-12-27/977943646888836.png' />证明在(0,1)内存在一点c,使得f'(c)=0.

设函数f(x)在区间[0,+∞)上连续、单调不减且f(0)≥0.试证函数 <img src='https://img2.soutiyun.com/ask/2020-12-13/976722177817809.png' /> 在[0,+∞)上连续且单调增加[其中n＞0].

设函数f(x)在[a,b]上连续,且f(x)＞0,证明:在(a,b)内存在一个ξ,使得 <img src='https://img2.soutiyun.com/ask/2021-01-14/979465674691464.png' />

设(x)在[a，b]上连续，在(a，b)内可导且f'(x)≤0， <img src='https://img2.soutiyun.com/ask/2020-08-06/965576645302938.png' /> 证明在(a，b)内有F'(x)≤0.

设f(x)在[a, b]上连续，在(a, b)内可导且f'(x)≤0, <img src='https://img2.soutiyun.com/ask/2020-12-14/976805726019948.png' /> 证明:在(a, b)内有F'(a)≤0

设函数f(x)在[0，1]上连续，且f(0)= f(1)，证明一定存在x∈(0,<img src='https://img2.soutiyun.com/ask/2020-12-20/977320815878019.png' />)使得f(x₀)= f(x₀+<img src='https://img2.soutiyun.com/ask/2020-12-20/977320902712985.png' />).

设函数f(x)和g(x)在[0,1]上有连续导数,且f(0)=0,f'(x)≥0,g'(x)≥0.证明:对任何a∈[0,1],都有 <img src='https://img2.soutiyun.com/ask/2020-12-13/97672399961901.png' />

设函数f(x)在闭区间[0,1]上连续,在开区间(0,1)上大于零,并满足 <img src='https://img2.soutiyun.com/ask/2020-12-16/976979475299148.png' /> 进一步,假设曲线y=f(x)与直线x=1和y=0所围的图形S的面积为2. (1)求函数f(x); (2)当a为何值时,图形S绕x轴旋转一周所得旋转体的体积最小？

设f(x)在[0，1]上连续，且0≤f(x)≤1，试证在[0，1]内至少存在—个ξ，使f(ξ)＝ξ.

设函数f(x)在区间[a,b]上连续,且f(x)≥0,那么<img src='https://img2.soutiyun.com/ask/2020-11-13/974109593488152.png' />(x)dx在几何上表示什么？

设f(x)在区间[a,b]上连续,g(x)在区间[a,b]上连续且不变号.证明至少存在一点 x<img src='https://img2.soutiyun.com/ask/2020-12-04/97592702964699.png' />[a,b],使下式成立 <img src='https://img2.soutiyun.com/ask/2020-12-04/975927090499471.png' />

设函数f(x)在[0,1]上连续,在(0,1)内可导,且<img src='https://img2.soutiyun.com/ask/2020-08-07/965639441738848.png' />证明在(0,1)内存在一点ξ,使f'(ξ)=0。

设函数<img src='https://img2.soutiyun.com/ask/2020-12-16/976976603992918.png' />,其中函数g(x)在(-∞,+∞)上连续,且 g(1)=5,<img src='https://img2.soutiyun.com/ask/2020-12-16/976976616554637.png' />,证明<img src='https://img2.soutiyun.com/ask/2020-12-16/976976676821084.png' />,并计算f''(1)和F'''(1).

设<img src='https://img2.soutiyun.com/ask/2021-01-28/980692750486118.png' /> (1)证明f(x)在[0,+∞)上可导，且一致连续; (2)证明反常积分<img src='https://img2.soutiyun.com/ask/2021-01-28/980692795149672.png' />发散。

设f在可求面积的区域D上连续.证明:若在D上(x,y)≥0,f(x,y)≠0,则<img src='https://img2.soutiyun.com/ask/2021-01-06/978805614690088.png' />

设函数f(x)在[01]上二阶可导,且f"(x)≤0,x∈[0,1],证明: <img src='https://img2.soutiyun.com/ask/2020-12-16/976976979900419.png' />

设f(x)在[a,b]上连续,在(a,b)内可导且f'(x)≤0, <img src='https://img2.soutiyun.com/ask/2020-12-04/975925572077622.png' /> 证明在(a,b)内有F'(x)＜0.