设一元函数f（u)在[-1,1]上连续，证明其中Ω为单位球。

A . A.AB B . DE C . CE D . CD

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f(x)在区间[0,1]上连续，证明:<img src='https://img2.soutiyun.com/ask/2020-12-15/976873792952469.png' />

设f(u)为连续函数,Ω(a)是半径为a的球体:x²+y²+z²≤2ay,求极限 <img src='https://img2.soutiyun.com/ask/2021-01-10/979150617925292.png' />

设f(z)在|z|< 1内解析,在|z|≤1上连续,试证: <img src='https://img2.soutiyun.com/ask/2020-08-06/965575689792285.png' /> 其中z属于C的内部.

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f(x)为[-a，a]上的连续函数，证明： (1)若f(x)是偶函数，则<img src='https://img2.soutiyun.com/ask/2020-10-12/971368555517115.png' />是[-a，a]上的奇函数; (2)若f(x)是奇函数，则<img src='https://img2.soutiyun.com/ask/2020-10-12/971368586317876.png' />是[-a，a]上的偶函数。

设f(x)为[0,1]上的非负单调非增连续函数(即当x＜y时,f(x)≥f(y)).利用积分中值 定理证明:对于0＜a＜β＜1.有下面的不等式成立 <img src='https://img2.soutiyun.com/ask/2020-11-30/975612485146551.png' />

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？

设函数f(u,v)在R²上具有二阶连续偏导数。证明:函数 <img src='https://img2.soutiyun.com/ask/2021-02-02/981107588858527.png' />

已知函数f(x)在[0，1]上连续，在(0，1)内可导，且f(0)＝0，f(1)＝1．证明： (I)存在ξ∈(0，1)，使得f(ξ)＝1－ξ； (Ⅱ)存在两个不同的点η，ζ∈(0，1)，使得fˊ(η)fˊ(ζ)＝1．

设函数<img src='https://img2.soutiyun.com/ask/2020-08-18/96660742963403.png' />其中g(x)有二阶连续导函数，且g(0)=1. (1)确定a的值，使f(x)在点x=0处连续； (2)求f'(x)； (3)讨论f'(x)在点x=0处的连续性.

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u(x,y,z)和v(x,y,z)在闭区域Ω上具有一阶及二阶连续偏导数，证明 <img src='https://img2.soutiyun.com/ask/2020-12-21/977438271832634.png' /> 其中<img src='https://img2.soutiyun.com/ask/2020-12-21/977437317641057.png' />是闭区域Ω的整个边界曲面，<img src='https://img2.soutiyun.com/ask/2020-12-21/977438285982443.png' />为函数v(x,y,z)沿<img src='https://img2.soutiyun.com/ask/2020-12-21/977437317641057.png' />的外法线方向的方向导数。这个公式叫做格林第一公式.

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' />

设函数f(x)在[a,b]上连续,a≤x₁＜x₂＜...＜x_n≤b,证明在[a,b]中必有ξ,使得 <img src='https://img2.soutiyun.com/ask/2020-12-15/976895957488208.png' />

设f(x)在[0,1]上连续且单调递减,则函数<img src='https://img2.soutiyun.com/ask/2020-12-11/976547106148916.png' />在(0,1)内(). A.单调增加 B.单调减少 C.有极大值 D.有极小值

设f(x)为连续函数，又<img src='https://img2.soutiyun.com/ask/2020-12-20/977330361227981.png' />， 证明: (1)若f(x)为奇函数，则F(x)为偶函数. (2) 若f(x)为偶函数，则F(x)为奇函数.